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See also: Instant Messaging - Chat in Social Networking

SMS (Short Message Service) Short (≤ 160 characters) text messages
MMS (Multimedia Messaging Service)
Device Required iOS Required Connectivity Required Recipient Types Allowed Content Types Allowed
SMS iPhone 1.0 and later Cellular Phone number Text only
MMS iPhone 3G and later 3.1 and later Cellular data Phone number Text, audio1, photos, and video
iMessage iPhone, iPad, or iPod touch 5.0 and later Cellular data or Wi-Fi Phone number or email address Text, audio1, photos, and video
Source: Communities discussions.apple.com

SMS (Short Message Service) is basically email for mobile phones also called texting. They can be sent phone to phone, or from a computer email system to a phone.
They are store and forward systems like email, where messages are saved until the recipient gets online and retrieves them.
It was designed in 1985 as part of the GSM (Global System for Mobile Communications) used primarily in Europe then. Phone screens were small to message size was limited to 160 characters including the header. When sent from an email system headers ranged from 20 to 30 characters leaving 130 characters for the message.
SMS text messaging is the most widely used data application in the world, with 2.4 billion active users, or 74% of all mobile phone subscribers.

See:
problems with broken up and out of sequence messages below.
Text Messaging – Help center | ting.com

MMS (Multimedia Messaging Service)
A standard way to send messages that include multimedia content to and from mobile phones. It extends the core SMS (Short Message Service) capability that allowed exchange of text messages only up to 160 characters in length.

The most popular use is to send photographs from camera-equipped handsets, although it is also popular as a method of delivering news and entertainment content including videos, pictures, text pages and ringtones.

MMS messages require a cellular data connection unless you are using a 3rd party app.
On Android and Windows phones you'll need to download a 3rd party app and the person you want to send messages to will also need to be using that app.

See: Multimedia Messages (MMS) FAQ – Help center

The standard is developed by the Open Mobile Alliance (OMA), although during development it was part of the 3GPP (3rd Generation Partnership Project) and WAP (Wireless Application Protocol) groups.

Although the standard does not specify a maximum size for a message, 300 kB is the current recommended size used by networks due to some limitations on the WAP gateway side

On Android and Windows phones you'll need to download a 3rd party app and the person you want to send messages to will also need to be using that app.

Instant Messaging (IM) is real time communication where participants have to be connected at the same time.
It uses push mode where the sender pushes messages to the recipient (assuming they have left their reader on) rather that a store and forward system in SMS where the recipient pulls their messages down.
These systems first used in time sharing systems used by universities and research organizations in the mid-1960s.
Internet Relay Chat (IRC) created in 1988 was used for group discussion forums.
AOL Instant Messenger (abbreviated AIM) was released in 1997 and was the dominant IM for 10 years. AN audio/video (A/V) chat program CU-SeeMe was developed by Cornell for the Macintosh in 1992.

The term "Instant Messenger" is a service mark of Time Warner and may not be used in software not affiliated with AOL in the United States.

Architectures - Protocols:


MSC - Mobile Switching Center - Switch data between users on a network
STP - Signaling Transfer Point
HLR - Home Location Register - Subscriber Information
VLR - Visitor Location Register - Roaming Subscriber
SMSC - SMS Center - Transfer the message to the MSC
                     that is serving the addressed mobile device.
SMPP - Short Message Peer to Peer Protocol
ESME - External Short Messaging Entities

Source: Communication Networks - SMS: How does it work?
Other A/V chat systems were introduced in the 90s as separate programs or Internet browser add ons. See IM - Chat at the Social Networking page. 
   - AIM Formerly AOL Instant Messenger
    - FaceTime (Apple) 
   - iMessage/Messages (Apple)  †
   - Hangouts (Google) * 
   - WhatsApp  (no message charges)
   - Viber  (no message charges)
   - BBM  
   - Facebook messenger  (no message charges)
 Old IMs:
  * Google Talk (Win)  and Google Chat 
  were replaced by Hangouts - Text Message, Phone, Video
  † Apple iChat was replaced by Messages
  See: 

 - 
Best Messaging Apps: WhatsApp vs Skype vs 
  BBM vs iMessage vs Hangouts | Digital Trends
 - 
Five apps as good as or better than iMessage
   in iOS 8 - CNET
   Skype
Internet Phone Services using VoIP (Voice over IP) proliferated in early 2000. Skype was a computer to computer calling service which expanded to computer to land line calling.

I use the same ID for my laptop and tablet and it will call both, if they are logged in.
They charge for calling to a land line, about 2¢ per min.

Broken up and out of sequence text messages appears to be a problem when sending between carriers e.g. Verizon to AT&T (Sprint is the worst). 
  It can happen if the message
Is too long. > 160 characters including header
Contains
  -  Characters in some languages, including Russian, Greek, Chinese,     -  Japanese, and Arabic.
  -  Letters with accents (å, é, ö, and so on).
  -  Non-ASCII symbols ( ● ¢ ¼ ).
  -  Emoji.
I think the long messages is more of a problem now than the special characters.
See support.apple.com

It may be a problem with Verizon. See forums.androidcentral.com/

Most of the posts on this are old 2011-13, so I don't know why it is still occurring in 2015. I'll try to keep my messages under 120 char.

Terms: 

EMS- Enterprise messaging system
SAR - Segmentation And Reassembly
WAP - Wireless Access Protocol
WTP - Wireless Transaction Protocol

Books:
Mobile Messaging Technologies and Services: SMS, EMS and MMS

Links:
IM - Chat at the Social Networking page
www.seattle.gov/tech/docs/GetOnline_SkypeTutorial.pdf
Definition of SMS Messaging - What is SMS Messaging? What is Text Messaging?
SMS - Wikipedia, the free encyclopedia
Instant Messaging - Chat Clients Talkabout - Google Blog about voice and IM
Social Networking

last updated 1 Oct 2013