Basically, they all fall under the generally category of "viruses".
However, there are a few distinctions.

Virus - Technically, a virus infects another file (attaches or inserts
itself into it). They usually infect program files or MS Office
documents. From there, it can replicate, do damage, etc. Unlike a worm,
these do not function as a stand alone (except possibly to infect a
given file).

Worm - This is almost identical to a "true virus", except that it lives
on its own and generally doesn't infect other files (although it can
replace them). Usually, these copy themselves using e-mail, networks,
disks, etc. Again, these are very close to a true virus, and can do the
same kind of damage.

Trojan Horse - This type of program doesn't copy itself but does do
damage to your computer. These types of programs rely on people to pass
them around and to run them. They do not e-mail themselves. The idea is
to make the program look like it's something harmless, like a screen
saver or joke, so it gets sent around.