Like other orchestral brass, the trombone has a mouthpiece, a main body of tubing, and a bell. The basically cylindrical shape of the trombone's tubing (as opposed to the more conical baritones and tubas) gives the trombone a clear, direct, brassy sound that is very popular in jazz and band music.
The instrument changes pitch using a moveable section of tubing called the slide. As the slide moves out, the instrument gets longer, and the sound gets lower. You might be tempted to think this means that there is one note available for any possible position of the trombone's slide, but this is not the case. Most possible placements of a trombone slide give pitches that sound wrong or out of tune, because they fall in between the notes of the chromatic scale. These in-between notes are only used when the trombone plays a glissando, sliding between the notes on purpose. There are seven slide positions that do give scale notes. Having the slide all the way in is position 1; having the slide all the way out is position 7. The other positions are spread out in between, with several inches between one position and the next.
But of course, the trombone can get more than seven notes. Like the brass instruments that only have a few valves (trumpet and horn, for example), the trombone can use changes in the player's embouchure to get many different notes from a different harmonic series at each position.
Seven positions, each a half step apart, will cover a tritone (about half an octave). This is plenty everywhere in the trombone's range, except at the very bottom, in between the fundamental and the second harmonic, which are a whole octave apart. So some trombones - especially bass trombones - have an extra length of tubing opened by a valve (called the plug or trigger) that allows them to play the rest of that lowest octave. (If you want or need to understand this paragraph, and don't, please see Harmonic Series.)
At one time, there was a complete family of trombones: a treble, which played in the same range as the modern trumpet, an alto, tenor, bass and contrabass. The instrument that is now commonly called "the trombone" is the tenor trombone. The trombone section of most orchestras and bands will also have a bass trombone, which has a deeper sound, a slightly lower range, and a fuller, more focussed sound on low notes. The only other trombone that is still played (although rarely) is the alto, which is smaller and higher than the tenor, but not as high as the treble.