The Clarinet family is quite a large family including the very low Contrabass Clarinet, the Bass Clarinet, the Alto Clarinet, the normal B flat Clarinet, and an E flat clarinet. When people say "Clarinet" it usually means the regular B flat Clarinet.
The Contrabass Clarinet is very optically impressive (2.7 metres!), but however is not critically important for sound. It is really only used for special effects, like giving an effect of extreme depth. It has an incredible range stretching almost three octaves. However, it is a big chore for maintenance. It is really big and hard to carry around, and your local Long and Mcquade music store won't have reeds for you (you can use Baritone Saxophone reeds though). You rarely see these Clarinets; you might catch sight of one in a big high school band.
The B flat Clarinet, or the Clarinet, is the most commonly used Clarinet out of the whole Clarinet family. Some call it the Soprano Clarinet, to fulfill the order of Bass, Alto, and Soprano. It is the oldest version; all Clarinets were based off of the original Clarinet. Some other Clarinets are toned in A, which means it would be one tone lower when both instruments play a note using the same fingering. A Clarinets are no longer used in Wind Orchestras or Jazz, but it is used in Classical music.
The E flat Clarinet is the smallest of the common Clarinets, and is noted in E flat. It produces a sharp and shrill sound, which makes it difficult to tune with other instruments in an orchestra. The E flat Clarinet looks as if a small child should play it- it is that small- but in reality it is extremely hard to play. Usually the E flat Clarinet is given to the player with the best ears, because you always have to be listening to see if you are in tune.