There are a number of ways to categorize headphones, but when most people talk about the different types, they generally mean the different ways of wearing 'phones.
Here I'll discuss all of the different types available and weigh up their advantages and disadvantages.
Full size headphones: Sometimes called circumaural, these are the classic "cans". Usually big and comfy, they go around the ears giving great sound quality. You do not see them in the home as often anymore, although some audiophiles swear by them. These 'phones remain an essential part of any recording studio and seem to be popular among those who want to listen to their iPods while looking like an extra from That 70's Show.
Ear pad headphones: Ear pad headphones are quite similar to full sized headphones but are smaller and sit on the ear rather than over it. In the trade, they are referred to as supra-aural and tend to have a slightly poorer sound quality than full sized 'phones but are a lot more portable due to their size. This style of headphones was very popular in the 80's but lost ground as earphones became more popular. They're now going through a renaissance, particularly due to the popularity of behind the head or earclip versions.
In-ear headphones: This type of headphone is gaining popularity due to their excellent sound quality and ability to block out external noise. They sit directly in the listener's ear canal forming a "seal" giving great sound isolation. Like anything, they do have a downside. Some users find them uncomfortable, especially after prolonged use, plus their excellent sound isolation does lead to a detachment from the sounds we are used to, particularly traffic!
Earphones: Earphones or earbuds are similar to in-ear headphones except they sit outside the ear canal rather than in it. They're generally the cheapest type of headphones but often give relatively poor sound quality. They are particularly popular at the moment with the white cabled iPod versions (and their copies) dominating the high street.
Earclip / behind the head headphones: Earclip and behind the head headphones are a bit of an oddity in this list. To be honest, these headphones could be any of the above types but either secure behind the ear or joined at the back of the head. So, for example, ear pad headphones that clip over the ear are now becoming much more popular than traditional over-the-head band headphones. Their popularity is quite recent and they gain a lot of use in the "sport headphones" market.