The standard clarinet is pitched in Bb - that is, when you finger and play what you read in your music as a C, the note which actually comes out is the Bb a tone lower. Most clarinet players, once they become more advanced, have a clarinet in A too - this is slightly longer and plays another semitone lower. For playing in certain keys it's more convenient to use an A clarinet rather than a Bb.
The C clarinet is the only clarinet where the note you play sounds at the written pitch - so it's very useful when playing music with others that doesn't include a designated clarinet (Bb) part - you can use it to replace a violin, oboe etc.
There's an Eb soprano, which is much shorter than the standard clarinet - it plays much higher, as one might expect, and they can be a bit wayward. There's one pitched in Ab, I think, which is really small and shrill, but it's rarely used.
The larger sizes are Eb alto, bass and contrabass (I'm not sure what keys the last two are in, as I don't have either of those!).