I split information on mattress into two pages.
1. This one with general information on construction methods, composition (chemicals etc.), polyurethane foam vs latex foam.
Contents: Considerations in selecting a mattress | Construction | Manufacturers/Price | Matress Size | Mattress Firmness | Indentation Load Deflection (ILD) | Types | | | Firmness | Problems with memory foam: | No air circulation: | Odors | toxic chemicals in memory foam mattresses | Green Mattresses | Sofa Mattresses | Sex Suitability: | Terms:

2. In products about specific mattresses - Specific brands, models and selection criteria.

Why a mattress is important | Considerations in selecting a mattress | Construction | Manufacturers/Prices | Types | Problems with memory foam | Mattress Firmness | Green Mattresses | Glossary of Terms

Why Mattresses are important:
A good mattress can remove pressure points on your body and lead to a better nights sleep so you have more energy throughout the day. Sleeping on the wrong mattress can cause or worsen lower back pain and put more strain on your heart.

Signs you need a new mattress:

• Your mattress is more than 10 years old.
• You just aren't sleeping comfortably anymore.
• You wake up achy and sore.
• You find that you actually sleep better away from home than you do in your own bed.
• Your mattress sags visibly, the spot where you sleep appears sunken in, or you can feel the hard wire springs underneath.

Latex:
Latex has a texture similar to memory foam, but it's springier, so your body will rest on the surface without sinking in. Most come with soft medium and firm. It sleeps cooler than memory foam.
Synthetic latex is petroleum based, and made from Styrene-Butadiene Rubber (SBR) which is the same molecular compound as natural latex.
While quite durable, synthetic latex is a bit less durable than natural latex.
Dunlop Vs. Talalay Latex Foam | SleepOnLatex.com
Natural Latex Vs. Synthetic Latex | plushBeds

Gel Foam
Gel is is a polymer made by blending plastics and other chemicals similar to memory foam.
It is used as both a standalone layer and an infusion into foam. Gel benefits mattresses by retaining less heat than memory foam. Also, newer gels compete against memory foam by providing better support, being more responsive and less toxic.

Copper and Graphice infusions:
Though gel is an effective and popular cooling agent, you may run across other infusions in your search for a mattress topper. Some others include graphite and copper, both of which work to absorb and dissipate body heat evenly throughout a structure. Copper infusions are especially interesting as there's some research to suggest that they're antimicrobial, which makes for an overall healthier sleeping experience.

Energex&tm; Foam
This hybrid provides the pressure relief of memory foam with the responsiveness and durability of latex.
This hybrid provides the pressure relief of memory foam with the responsiveness and durability of latex.
Energex VS. Memory Foam - YouTube

Plant-based memory foam
Plant-based memory foam is a open cell design that dissipates heat at a rate that is 25 percent faster than gel-infused products.
Amerisleep's Bio-Pur® is made from partially plant-based sources using a zero-emission process. That means it's the most eco-friendly material of its kind.
They are healthier and more environmentally-responsible Organic Mattress Labels You Can Trust - Consumer Reports

They can be coil or foam.
Coil
Coil is the most common type. They’re usually more affordable.
They are breathable and hold their shape nicely.
Coil mattresses are generally not very comfortable. There is typically less than an inch of padding over the coils, and the mattress itself isn’t very thick, so it doesn’t provide much protection from the sleeper mechanism.

Foam and Polyester:
For the most straightforward futon mattress, look for one constructed of layers of foam and polyester. These mattresses offer a bit less support than other options, but they tend to be pretty affordable and lightweight Advantages and Disadvantages of Sleeper Sofa Mattress Types The Difference Between Sofa Bed Mattresses and Regular Mattresses - Lully Sleep

References:
Memory Foam: Traditional, Gel, or Plant-Based? | Simply Rest
Buyer Beware: Gel Memory Foam Mattress
in Home and Garden

Aggregates 10,000s of mattress reviews, product reviews, and independent scores on the net. With this wealth of data, we created weighted scores on over 100 different buyer dimensions.
They give only 1 score for manufactures with several different types of toppers, so I think their scores may be somewhat arbitrary.

Construction

Mattresses are generally composed of two main sections
1. A support section on the bottom
2. A comfort section on the Top 
3. A fabric cover (ticking)  The covering over comfort section
 may have some padding also.
Foundation - A box spring or platform that the whole thing sits on.
 Usually 8 inches.
  
There are several types:
Support
  Innerspring - Wire coils or interconnected springs
  Foam (Polyurethane or latex) 
  Adjustable air like the Sleep Number beds
  

One of the more complex mattresses is the Summerfield Geneva

Comfort Section:
  1/2 Ounce Fr Barrier 
  1" Poly Foam x 2
  1/2" Gel Foam Center Third 
  7/8" High Density Foam
  1.25" Compressed Poly Board 
  1.5" Insulator Pad
Support Section
 Edge Support System
  12 3/4 Gauge Coil System 

Sealy Posturepedic Hybrid Trust Cushion - $1,275 (Top rated at Consumer Reports)

Novaform 14" Serafina - $1,575

Mattress thickness: 14"
Top layer: 3" of gel memory foam for cool, cradling comfort
Middle layer: 3" of air channel foam for enhanced air flow
Base layer: 8" of base foam for long-lasting support
Patented gel-infused memory foam technology
Cover: Quilted, plush cover
Mattress cover is not removable for washing; spot clean only;
   follow all manufacturers' care and use instructions

Top Pad - 

Support
  3,288 interconnected springs (queen)

Matress Size - sheet sizes

                 Mattress   Flat Sheet Sizes:  Comforters/Duvets:
                 W       L
 Twin or single  38-39" x 74"   66 x 96"          66 x 90"
 Full or double  53-54" x 75"   81 x 96"          76 x 90"
 Queen           60"    x 80"   90 x 102"         90 x 94" 
 Short Queen     60" x 72"     (Sofa bed, Sleeper sofa)
 Olympic Queen   66" x 80"     (Extra Wide Queen)
 King            75-76" x 80"  108 x 102"        104 x 94"
 California king, 72"   x 84"  108 x 102"
 
 Note: Fitted sheet sizes are the same as mattress size.
 See  See Bedding Guide - Sheet size | blair.com

Thickness

New mattresses designs add extra padding and have greater thickness.  
This will require deep pocket sheets to fit correctly.

                 Better Homes & 
                 Gardens 2017    Hunker 2018
* STANDARD DEPTH: 7 to 9 inches       7-12"
*  DEEP MATTRESS: 10 to 15 inches    13-17"
*     EXTRA DEEP: 16 to 22 inches    18-25"
Deep Pocket Sheets are required for mattresses over 14" deep.
Sheet pocket sizes
7 inches to 14 inches (standard),
15 inches (deep),
and 16 inches to 22 inches (extra deep). 
 

Back Pain:
Traditional wisdom was that a firm mattress was best for your back. This was when soft mattresses allowed sag so your back was curved. What you want is a mattress which allows you back to rest in a normal position. You want it to avoid sag but to be soft enough for your shoulders and hips to sink in when on your side. New mattresses use things like independent coils, multiple density layering and memory foam to avoid sag and still have form fitting capabilities. A medium firm mattress is usually best.
If you sleep on your side you need a softer mattress to allow your shoulders and hips to sink in. If you sleep on your back (better for people with back problems) a firmer mattress will work.

There are pros and cons to waterbeds. If you lie on your stomach, the water bed allows you to increase the sway or the sag of the back.
Claims that a mattress is "orthopedic" or "medically-approved" should be viewed skeptically. There has not been extensive medical research or controlled clinical trials on the topic of mattresses and low back pain.
See:
Choosing the Best Mattress for Lower Back Pain - spine-health.com
Mattresses for Back Pain - Back Pain Center - EverydayHealth.com
Supportive Bedding - back-shop.com
The AlluraBed by Tempur-Pedic - The Healthy Back Store

Indentation Load Deflection (ILD) (ILD - Pounds to compress a 4" thick sample to 3" (25%))

• Soft 20-24
• Medium. 25-29 Recommended for those who enjoy a softer feel.
• Regular firm. 30-34 Recommended for those who like a good firm feel, but not extra firm. Usually recommended for those who weigh less that 150 pounds.
• Extra firm. 35-39 Recommended for those who like extra firm support.
• Ultra firm*. 40-44 Recommended only for those who insist that they enjoy sleeping on the floor.

Foam Density, the weight of a 12" x 12" x 12" block, is related
The relation between density and ILD varies depending on materials.
Density of foam reflects the quality and durability of the material more than the feel.
A high quality foam will be denser for the same ILD.
At Foam Factory they have:
1.8 lb/ft³ density ≅ 35 lb ILD
1.8 lb/ft³ Lux foam ≅ 50lb ILD
2.8 High Quality (HQ) ≅ 35 lb ILD
2.8 Lux Foam (HQ) ≅ 50 lb ILD
3 LB memory foam ≅ ILD 14
See The Differences and Relationship Between Foam's Density, Weight, and Firmness | The Foam Factory

Consumer Reports found that most mattresses advertised as firm were actually Medium or Soft.

Innerspring Mattresses
Innerspring mattresses have a support layer of wire springs. Either a continuous coil or individual coils in a variety of configurations.
Individual coils can be either straight (Marshal coil and LFK Coil) or an hourglass shape (Bonnell Coil and Offset Coil). Individual coils are frequently wrapped in fabric (pocket coils)

In Consumer Report tests, a higher coil count or firmer mattress didn't necessairly reduce sagging.

See: The Best Mattress For You - Support - The Original Mattress Factory
and coils Leggett & Platt Components

See:
Latex Foam Rubber and Flexible Polyurethane Foam (FPF) at Polyurethane Foam Association (PFA)
Commercial Foam - Insulation, Packaging, Closed-Cell Foam & More

Memory Foam:
Memory foam is essentially made from the same chemicals as polyurethane foam, but it has added chemicals to it that change the way it feels.

Some companies make what they call natural memory foam.

The Denver Mattress Company makes Organic HeveaPUR^® latex is resilient, temperature moderating, and naturally antimicrobial to reduce pressure points while you sleep, year after year. HeveaPUR^® sleep system! Derived from the Hevea Brasiliensis tree.
There iChoice^® is topped with 3" of HeveaPUR.

Essetia has what they call a natural memory foam made with latex and other natural ingredients.
see: Where & How Essentia Mattresses are Made

Keetsa replaces 12% of the petroleum in their memory foam with castor oil.

Others solve the heat dissipation problem of memory foam in various ways.
Sealy has OptiCool^® Gel Memory Foam.
Sealy also has OptiCool^® Gel Latex

Foam - All foams are basically open-cell.
Polyurethane and latex foam.

Density - Technically the weight per unit of volume (specific gravity). Density is commonly misused to mean the hardness or softness of a material, specifically in our case, sponge. Many times someone will ask "What is the density of the material?" What they really mean, in most cases, is "What is the hardness or softness?" The correct term for the hardness or softness of a piece of cellular rubber is compression deflection.

Compression Deflection - Correct Measure of hardness. The weight required to compress a 1 cu. in. cube 25%. Different companies use different kinds of Visco Elastic foam also called memory foam, slow recovery or Swiss Foam. They seem to be based on the foam developed by NASA (see below), but I haven't seen any comparisons. e.g. Isotonic uses something called Conform and ViscoLux. HealthyBack uses a foam from Obus Form

Visco-Elastic Memory Foam (memory foam):
From abed.com

Developed for NASA to relieve extreme G-force pressures. Memory Foam molds to your body contours to help prevent tossing and turning during the night. When you lie down on this amazing material, the heat and pressure sensitive foam reacts to your body's weight and temperature, so that it molds to your exact body shape. This means that whether you sleep on your back, stomach or side, your weight is evenly distributed and your spine remains in a neutral position.

Tests have shown that these mattresses actually improve sleep and increase the level of deep sleep, so you wake up more alert, refreshed and better rested. Recommended by over 25,000 medical professionals worldwide. Naturally treated cover protects you from harmful allergens, resists dust mites, and removes easily for washing.

Problems with memory foam:
There are 3 problems with most memory foam mattresses: no air circulation, Odors and toxic chemicals.
Note: Some memory foam mattresses claim to eliminate these thru various means; latex, different fire retardants, ...

No air circulation:
One problem reported with these mattresses is they do not breath and can be too warm in the summer. I got the following response from abed.com regarding this:

From: "Roseanne McArdle, sleep consultant" <Roseanne@abed.com>
To answer your question regarding humidity, what the article is referring to is the need to sleep on a bed that does not make you sweat. Our foam beds such as the Dormia and the Wellington have Joma wool filling quilted into their covers and this allows for the air circulation your bed needs to have to help regulate your body temperature and keep you from overheating. Latex foam also is well known for helping air to circulate and keeping heat away.

Odors:
About 15% of people who buy a memory foam mattress report a noticeable odor. For most people, this is a just a minor annoyance that dissipates within a few days to a few weeks and accounts for less than 2% of returns according to Sleep Like The Dead. However, a small percentage of people may report respiratory irritation and other symptoms like nausea and headaches.

In 2007 mattresses were required to be fire retardant.
Around then there were reports of toxic chemicals in memory foam mattresses and a class action law suit was filed.
Virtually all commercially available mattresses made today are filled with petroleum based foam and then soaked in a pound or more of fire retardant chemicals. These mattresses outgas formaldehyde as they age, which is a known carcinogen. Fire redardants, made of chemicals called Polybrominated diphenyl ethers (PBDEs), build up in our bodies like the banned PCBs and have been associated with neurological damage and cancer."
Source: Organic Mattresses | DAXstores.com
See:
Toxic Mattress Chemicals | Are Memory Foam Mattresses Safe?
Should You Ditch Your Chemical Mattress? | Mother Jones
CR4 - Thread: Memory Foam or Foamy Memory?
Chemicals in Mattresses and Pillows Linked to Many Illness
Rock-a-bye baby, chemicals and all: the health risks posed by crib mattresses | National Center For Health Research
Memory Foam Mattresses: Benefits and Disadvantages | WebMDd

Note: The amount of formaldehyde emitted by a mattress is probably less than that in normal homes which comes from particle board, carpeting, paints, even your comuter. But it concerns many because you are spending 8 hours a day with your face right next to it.
See En Update On Formaldehyda - Consumer Product Safety Commission

Comforpedic by Simmons uses a new advanced level of memory foam called, NXG. In its manufacturing process, the foam is created in a VPF (variable pressure chamber) which removes the notorious, yet, unavoidable chemical irritants from the product.

Latex Foam:
From: abed.com Derived from the have brasiliensis, or the rubber tree, latex is the most naturally durable cushioning material available. Latex is harvested in much the same way as maple syrup from the maple tree. The milky white latex is converted into a mattress or pillow in special molds and undergoes four rinsing processes to eliminate odors.

Health Benefits Latex is antimicrobial and dust mite resistant, making it perfect for pulmonary allergy sufferers. It's ecologically-friendly and has no chemical properties or detrimental impacts to the environment.

Additional Benefits Designed for breathability with an open-cell structure and special "pin core" construction, latex allows for ventilation of body heat. It also keeps moisture from building up next to your body. Aluminum core pins transfer curing heat into the foam. As a result, thousands of "pin-core" air-channels enable natural ventilation of the final product. Types of Latex:
Talalay latex - Has less resistance, bouncier, and is more for soreness or those who have or seek relief from pressure point problems.
Dunlop - When all else is equal dunlop is firmer, denser, and may provide extra needed support.

Some companies allow you to select the material for each layer. Some people will combine Talaly and dunlop in different layers.

Blended latex will have a percentage of natural latex mixed with a petroleum based material called SBR that will off gas.

Latex vs Polyurethane Foam:

Latex lasts longer up to 20 years.
More exact pinpointed support than memory foam. 
Does not release the harmful chemicals
Made with a much cleaner and less energy intense process
Latex is more expensive

See:
Choosing between Polyurethane Foam, Memory Foam and Latex Foam | SleepOnLatex.com
Latex Mattress vs Memory Foam Mattress - Latex versus Foam Mattress Comparison

Covers:
The elasticity of stretch knit cotton facilitates contour while reducing surface tension, so the mattress material can do its job..

Spring vs Foam:
There are conflicting report on which is best.
Note: We are talking about the support layer here. Most innerspring mattress have foam comfort layers on top.

In consumer reports 2015 tests innerspring mattress got marginally higher scores.
However, foam mattresses seem to get the nod when you search for the question online.

In trying mattresses recently, I found the spring mattresses with independent coils slightly better.
This makes sense to me because foam mattresses are like connected spring mattresses, when you push down on one spot it tends to pull the neighboring region down, because they are attached. Independent coils are not connected and don't do this.

Spring mattresses will make it easier for heavy people to get out of bed.

At Spring Mattress vs Memory Foam Mattress -| Essentia they claim foam mattresses give you better pressure relief among other things.

At What is better, a foam mattress or a spring mattress? - Quora they have contradictory reports.

At Sleep Like The Dead they say
"Innerspring beds overall have lower owner satisfaction than most other mattress types due mainly to below-average durability and longevity. They also tend to provide only fair long-term pain and pressure-point relief. Some models may produce noise."

Green mattresses use natural latex harvested from rubber trees for their foam. It is more durable and the production process is much cleaner than that from polyurethane foam. It is also more breathable and less flammable.

Petroleum foam produces harmful gasses (which dissipate over time) and have a fire retardant chemical, which is toxic, added.

Beware of what some manufacturers call "organic mattresses", they may have some organic cotton, but that's all.

Nest advertises the Love Bed as "A bed engineered with your love life and your sleep life in mind."
It is all foam - 3" Latex top and 6\0x2033 CertiPUR high density foam support layer

Be thinking about how it feels on your pressure points (your head, neck and shoulders, hips and heals). Does it give good support for the curve of your spine? Is it too firm or too cushy? How does it feel when you roll over?

Tryouts aren't usually an option at places like Costco and BJs, so checking return policies before you buy is extra important.

It can take 30-45 days to adjust to a new sleeping surface. It is not unusual to take some time to get used to a new mattress and experience some soreness..

Companies:
Memory Foam Mattress Brands: Ratings and Comparison

Essentia Organic Mattresses & Natural Memory Foam Mattresses
Essetia has what they call a natural memory foam made with latex and other natural ingredients.
see: Where & How Essentia Mattresses are Made

Naturepedic Natural & Organic Mattress, Luxury Organic Mattress & Organic Bedding
Organic Mattresses, Natural Latex Mattresses, and Organic Bedding | Savvy Rest
Dealers
Green Mattress Stores in the SF bay area:
Essentia - 1799-D Fourth St, Berkeley
Keetsa - 68 Division Street, San Francisco
The Natural Mattress Store - 1243 Sutter St, San Francisco
Nest bedding - Nest 367 9th st, San Francisco

Which Natural Mattress is Right For You? | Mama Natural

Others:

Organic Mattresses - Toxin Free Natural Latex | stlbeds.com
My green mattress 
Coyuchi 
Organic Mattresses | Dax Stores

Manufactures
Duxiana
Tempur-pedic - Brookstone
FloBeds Latex
Foam-Fabrication.com
FoamByMail.com
Select Comfort
ComfortDirect Air Bed
Simmons
Sealy
Serta

Terms:

• ASTM - American Society of the International Association for Testing and Materials
• CertiPUR^® A voluntary testing, analysis and certification program developed by members of the global foam industry in response to an increase in substandard imported foam products. Made without harmful chemicals.
• FPF - Flexible Polyurethane Foam
• GOLS -
• GOTS - Global Organic Textile Standard
• IFD - Indentation Force Deflection - Same as ILD for rubber.
• ILD Indentation Load Deflection
    Pounds to compress a 4" thick sample to 3" (25%))
  See ILD above
• PBDE - Polybrominated diphenyl ethers - A harmful fire retardent used in some foams.
  
• VOC - Volatile Organic Compound

See Also:
See mattresses in products for specific brands, models and selection criteria.
Berkeley Parents Network Recommendations
Mattress Reviews on Latex Mattress and Spring Mattress | Mattress Online
Mattresses and sleep
Mattress Review: Best Mattresses - ConsumerSearch
Mattresses and Sleep, a pseudo-technical discussion of sleep by a company selling memory foam mattresses
Back Pain and Injuries.
2018 Best Mattress Video - tip about mattresses from 13 year expert - YouTube 1:10