Noise Canceling
dB ratings are claims by manufacturers. I haven't found an independent test yet.
Bose
Panasonic
300g 18 dB $200
150g 12 dB $100
Shure:
ear buds 37 dB

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 Ð Digital Analog Converters Ð DACs | 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.