Alcove tubs with apron - these are the most common. These tubs are meant for both bathing and showering in. They sit in between 3 walls and usually have some kind of waterproof surround like tile. To install these tubs, you don't need to build a "deck" or a front apron (i.e. the front vertical face of the tub) because the tub comes with an integral apron. The most important thing about these tubs is that they have raised tiling flanges on the three sides that go up against the wall. The tiling flange is a lip that sits behind the cement backer board and prevents water from getting behind the surround and leaking into the room below. Below is the tub we've chosen for it's clean lines, it is an Azzura Adora, and shown is the alcove version.
Drop-in tubs - these are more commonly used in larger bathrooms where there is room for a separate shower stall. These tubs do not have tile flanges and are meant for bathing only. With these you do need to build a deck to support the tub.
Alcove without apron - these are relatively uncommon. They are basically the same as the alcove described above, except you need to build a front apron/deck on the front. These tubs are also meant for bathing and showering, but give the owner the option to finish the front in a material like tile or panelling.
Free-standing tubs - These are meant for bathing in only. And in a lot of cases, they're actually meant for looks only, because most people would be too afraid to dirty them. These are the tubs for the rich and famous...without children...with staff. They start around $1,000 and go up to the stratoshpere. I've seen $25,000 copper tubs, $30,000 marble tubs, etc. Better make sure you get a really good plumber to install yours for you!