The Tenor Trombone
The tenor trombone is pitched in B flat (i.e. it sounds a tone lower than music written for it in 'C', and its bottom note is B flat). This is the instrument that is most often used in classical and jazz situations. Orchestral models tend to have a larger bore size, while most jazz players prefer the brighter sound of small-bore trombones.
The Alto Trombone
The alto trombone is pitched a fourth higher than the tenor, in E flat. It's used primarily in specific orchestral situations, and with its shorter scale than the tenor, tuning can be a little tricky as the slide positions are closer together. Its sound is brighter and clearer, which can be very effective in a recording studio. However it's a little used instrument, partly due to the fact that classical composers write for it in the alto clef, which most players don't read.
The Bass Trombone
The bass trombone is pitched in B flat as the tenor, but usually has a wider bore to give a fuller, richer sound. In order to give the instrument a seamless range from the 'pedal tones' upward (notes not normally accessible on a tenor trombone), the bass trombone is usually equipped with either one or two rotary valves which bring additional lengths of tubing into play. These valves effectively lower the bass trombone's pitch to F, E flat and sometimes G.
The Valve Trombone
The valve trombone generally has no slide, but instead three valves similar to those of a trumpet or tuba, and is most often used in jazz or big band music by soloists - often trumpeters choosing to 'double' on the instrument. The 'superbone' is a rarer version that has both valves and a slide. Because of the extended intervals involved, the valve trombone can be hard to play in tune.
The Soprano Trombone
The soprano trombone is pitched an octave higher than a tenor trombone, and is sometimes called a 'slide trumpet'. It's very rarely heard as its small dimensions also make it very hard to play in tune.