The clarinet is a very versatile instrument, with different shapes and tunings, like other woodwinds. Probably, though, the clarinet is the woodwind that has more variations. However most of these types of clarinets are not widely used, nor is there much literature for them.
The main advantage for clarinet players is that all clarinets share the same fingering (with little variations) in the same system. I.e., all Boehm clarinets share fingering, so if you know the notes on a soprano clarinet you will be able to know the notes on a Eb clarinet, Alto, Bass, etc. There will be some differences in playing techniques, though. As a probably too simple guide, the highest the instrument is, the more pressure I'll have to exert on the wind column. Also, the lowest the instrument is, the more volume of air I'll have to use.
These are the different types of clarinets from smallest to biggest.

Requinto

It is the smallest clarinet. It is a high-ptched instrument especially used in bands and clarinet choirs. Its use in orchestras is very limited and there isn't as many literature for it as there is for Bb soprano clarinets.
It is mainly an instrument devised for playing high notes, making it reasonably difficult to play in the chalumeau registry.
For a clarinetist who doubles (i.e., plays both the requinto and a soprano clarinet) it has an extra difficulty, because the keys and holes are closer (the instrument is smaller) and fingers have to be very close. .
Requintos are normally pitched in Eb, there are also rare requintos in Ab (called piccolos) and D.

Soprano clarinet

It is the most usual by far. When we speak about a clarinet, the first image that comes to mind is the soprano clarinet, which we simply call clarinet. For other types we specify (requinto, alto, etc). It is normally tuned in Bb (by far the most common) and A. A clarinets are normally used for orchestral music (18th and 19th centuries, mainly). For professional musicians it is advisable to have the pair (i.e. a Bb and an A clarinet) so they can play most of the repertoire for clarinet. If we think of the greatest concertos for clarinet and orchestra we will see that this is the case.

Example: Mozart's K622 Clarinet Concerto (A Clarinet or A Bassett Clarinet)
1st and 2nd Clarinet Concertos and Concertino Op 26 for clarinet and orchestra by Weber (Bb clarinet)
Poulenc's sonata (Bb clarinet)
And a long etc.

We can't forget the less usual tunings for clarinets, the ones known as clarinettes d'harmonie, tuned in C, D and G.

Bassett Clarinets

Bassett clarinets are clarinets evolved like corni di bassetto. They differ from soprano clarinets in the fact that they can reach a low C (conventionals clarinets reach E or Eb).

The most common tuning for bassett clarinets is A, though you can also find them in Bb and C

It is crucial not to indentify bassett clarinets with corni di bassetto. The Corno di bassetto is an instrument that evolved differently from the bassett clarinet, and it is tuned in F. Though they are similar, they are not the same instrument.

Alto Clarinet

It is also called Tenor clarinet. Created in the 19th Century and tuned in Eb, it is longer than the bassett clarinet, and its bell and neck confer it a shape closer to a saxophone.
We normally use altos for bands and clarinet choirs, its use for orchestras being close to non-existant (one of the few scores where it is used is Stravinsky's Threni).
Because to its pitch and sound, especially in the high registry, it is used as a substitute for alto saxophones in chamber music formations.
This clarinet is becoming gradually abandoned, since modern orchestrations for wind orchestras and fanfares don't include parts for alto clarinet. Bass clarinets and Alto Saxophones cover the space Alto Clarinets should occupy.

Bass Clarinet

It is very similar in shape to an alto clarinet but it is much bigger. It is widely used in clarinet choirs, orchestras, bands and fanfares and it has (comparing it to the other clarinet types except the sopranos) a lot of repertoire. However, many bass clarinetits use to play bassoon works to complete the literature available for their instrument.
Usually, bass clarinets are tuned in Bb, but there are also bass clarinets in A and C. There are two models of bass clarinet (the same goes for contralto and Contrabass clarinets): the one which reaches lower Eb and the one reaching C.
Bass clarinets have also been played outside "classical" mussic, for instance for jazz and pop (like The Beatles' When I'm Sixty-four).

Contralto Clarinet

Bigger (and with a lower pitch) than the bass clarinet, the Eb Contralto (or EEb Contralto, as it is known) is very similar to the contrabass clarinet. It is used for the same the contrabass clarinet is used, so we'll cover its uses in the following section.

Contrabass Clarinet

It is the biggest regularly available clarinet. Unlike contralto clarinets, contrabass clarinets are hard to find. The main makers (Selmer and Buffet) sell their contrabasses for astonishingly high prices (47,000 USD or 23,600 GBP).
Tuned in Bb (sometimes in C) it is a "pedal" instrument. This means that it is used for doubling tubas, baritones, contrabasses or to play the bass part of a melody in a band, fanfares, clarinet choirs, etc.
Because of its size, price and the fact that they are not easy to play in the middle and upper registry combined with the difficulty to find repertoire and accessories for the contrabass clarinet, it is becoming less and less used.
As with the contralto and the soprano, you can find contrabass clarinets made of metal.

Wooden


Metal

Experimental

"Experimental" clarinets are those which have been devised by luthiers but not mass-produced or widely used.

We will focus on three:
The quarter-tone clarinet: made by the German luthier Fritz Sch├╝ller, it was created to play quarter-tone melodies through two bores (tuned at a quarter-tone distance one from the other) joined to one mouthpiece. It is a clarinet which is unbelievably difficult to play.

L'OctoContralto
As its noun shows, it is a clarinet able to play one octave lower that the contralto. Only three of its kind were ever made by G.Leblanc, seen in a picure next to these lines playing it. .

L'OctoContrabass
There's only one Octocontrabass clarinet, made by Leblanc and registred in the Guinness book of records as the orchestral instrument able to play the lowest note (a near infrasound called Bb-1) Despite the fact that there was only one of its kind ever made, there are three works for octocontrabass clarinet and orchestra.

Quarter-tone
Octocontralto