The "Normal" Clarinet
The A clarinet and the B flat clarinet are the "normal" clarinets. They are the ones that you usually think of when you talk about "the" clarinet. Sometimes this clarinet is referred to as "soprano clarinet" (which is correct, thinking of clarinets resembling the voices Bass, Alto and Soprano).
B flat - by far the most frequently used
Of the two soprano clarinets the B flat clarinet is by far the most frequently used instrument - both in the wind orchestra and in jazz there are no A clarinets any more. The A clarinet is still widely used in classical music. The classical clarinet player carries about a set-case containing both instruments. There are many pieces where you have to use them both, depending on the key of the part that is played.
What Are A Clarinets Good For?
Of course one could - theoretically - transpose the notes, you probably would prefer to write the notes down. But in reality that turns out to become very difficult for the player, since most keys and a lot of jumps can't be executed so easily and as result the piece would sound much poorer than it would have to. If you plan to play the famous clarinet concerto in A by Mozart on a B flat clarinet, that will translate into B (Si) - which means five sharps (as opposed to none for the A clarinet).
Very handy: One Bore Diameter - One Mouthpiece
The A and B flat clarinet are very similar in size (only half a tone apart) and both have the same bore diameter - so you will use only one mouth piece for both instruments. That means that you can quickly change from A to B flat having a warm mouth piece with an already played-in reed, considerably reducing the risk of squeaking. One thing less to worry about! And of course you don't have to buy a second mouth piece and different reeds.
Despite its name the modern Bassett Clarinet is not a Bassett Horn but rather a soprano A clarinet (very rarely a B flat) that was extended by about 18 cm towards the bell and four extra keys. That means the tone range is extended four half tones (E flat to C). So it becomes possible to play Mozart's clarinet concerto KV 622 in the original form without the transposed parts (the lowest notes were transposed up by an octave to play it on a standard A clarinet). Today's professionals use this instrument when performing the concerto, you can watch Sharon Kam on this YouTube Video . With some professional type clarinets (high end / high price) you can buy a longer lower part for the instrument and it fits out-of-the-box, otherwise you can have it made custom-built for your favourite instrument.