Noise Canceling
2016 update:
You should never use noise-canceling while driving, riding a bike or walking in traffic or near train crossings.

Model	Price	Blue tooth	in/over/ on ear	CR			WSJ		mike	Cnet
				score	Noise	Sound quality	dB canc.	sound qual
Bose Quiet Comfort 35	$350.00	y	over				20.5	4	y	8.8
Sony h.ear MDR-100ABN	$350.00	y	over	86	5	5	17.1	4.5	y	8.6
Samsung Level Over	$260.00	y	over	80	4	4			y
Samsung Level On	$180.00	y	on				8.8
Phiaton BT220 NC	$160.00	y	in	76	3	5
Bose QuietComfort 25	$300.00	n	over	75	5	4			y	8.5
Beats by dre Studio Wireless	$250.00	y	over	72	4	4	8.0	3.5	y	8.3
Parrot Zik 3	$400.00	y	over	69	5	4	13.1		y	8.1
Koss QZ900	$200.00		over	66	4	4			n
Band & Olufsen BeoPlay H8	$500.00	y	on	64	5	4	12.0		y
Sony MDR-ZX770BN	$230.00	y	over	62	4	4			y	7.8
Sennheiser Momentum	$400.00	y	over	62	4	4	11.9	5		8.4
Sennheiser PXC 550 Wireless										8.3

WSJ - Mute Your Commute: The Best Noise-Canceling Headphones - WSJ June, 2016

CR - Noise-canceling headphone Ratings | Consumer Reports (69 and above recommended)
5 = excellent; 4 = very good; 3 = good

in-ear models will cancel some noise even without noise cancelling technology.

Comfort:
  On long airplane flights I found in-ear headphones caused my ears to itch 
  after a couple of hours.
  On-ear models made my ear sore after a couple of hours.
  My inexpensive over-ear model also made my ear sore, but supposedly
  the better over-ear models are better.
  

NY Times David Pogue June 14, 2007
The "active noise reduction" technology used by these headphones generally cancels only lower frequencies. Higher ones are difficult to stifle electronically. Whatever upper-register noise cancellation you get from these headphones, therefore, comes from the seal they make with your ears, which is "passive noise reduction."
Not as good as Bose but very good for the price. All but the highest frequencies are subtracted.
Audio-Technica ATH-ANC7 $132
Audio-Technica ATH-ANCI in-ear
Sony MDR-NC500D 14dB $400
Panasonic RP-HC500 $100
Bluetooth:
Earbuds, In-ear monitors (IEM):
Earbuds rest on the area outside the ear canal or may go in a little but do not rely on the ear canal to be held in pace. They are non-isolating; i.e. you can hear more outside noise.
They are better if you need to hear outside noise, for example if jogging on a road or when skiing where you need to hear someone yell "look out" before they crash into you.

In-ear models fit into and 'seal' the ear canal, much like standard earplugs.
According to HeadRoom, "These earphones can sound significantly better than traditional headphones or earbuds at the same price." These are sometimes called noise isolating, because they block more outside noise.

There are also mic'd headsets have a built in microphone for use with a smartphone.

Early iPods did not have very good amplifiers (I'm not sure of the current versions), and a headphone amplifier greatly improved the sound.
See:
Amplifiers - Digital Analog Converters - | HeadRoom Audio
Can a portable amplifier improve an iPod's sound? | The Audiophiliac - CNET News

Consumer reports lumps these in a category called portable headphones which includes some small over the ear models.

                        Score
Polk Audio Nue Era    $90   80  (integrated microphone)
Shure SE112           $50   71
Panasonic RP-TCM125   $10   66
Wireless
JBL by Harman J46BT   $80   71  ( integrated microphone)
Jabra Rox             $100      earbud (microphone)
Sol Republic Shadow Wireless  $100  c|net 4