A and B flat clarinets
The "normal" or soprano 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.
Why are there still A clarinets - what are they good for?
The A clarinet sounds a semitone lower than the Bb clarinet. In practical terms, this means: there is a C in the notes, you use the fingering for a C, play, and we hear an "A". With the B flat clarinet, we would hear a B flat. The notes for clarinettists are already transposed for the correct instrument. In pieces with many flat accidentals, the B-flat clarinet has two fewer Bs than a flute, and in pieces with many sharps, the A clarinet has three fewer sharps when used. This means that in pieces with B-flat keys, the B-flat clarinet is the better choice, and in pieces with lots of sharps (#), the A clarinet is the better choice.
Of course one could - at least theoretically - transpose the notes, but you probably would prefer to write the notes down. In reality it turns out to become very difficult for the player, since most keys and a lot of jumps can't be executed that easily and as a 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 would translate into B major - which means five sharps (as opposed to none for the A clarinet). The Bb clarinettist has to cope with these five crosses and unfortunate fingerings. And despite a lot of practising: the piece is difficult enough anyway, and with increased difficulty it just doesn't sound as good any more.
Classical pieces often have different keys in different movements. Then, classical clarinettists would switch instruments, depending on the key of the section that needs to be played. The classical composer has already provided for this, and in the notes it says "clar in A" or "clar in B" or just "in A", sometimes just by hand. If you are playing in a new orchestra for the first time - for example, as a substitute - it is better to ask beforehand and to look at the sheet music carefully. It's also best to look at the score to see if it's not better to bring both instruments and assemble play on them a bit before the rehearsal starts. If there is no other clue, the quickest way to recognize the keys is to compare the accidentals - the trumpets (also Bb instruments) must have the same number of accidentals.
In wind orchestras, pop and jazz - as I said - the soprano clarinet is almost always a Bb clarinet anyway.
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.
The modern bassett clarinet
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.