A saxophone is a single-reed woodwind, closely related to the clarinet. In fact, the mouthpieces are very similar, and many instrumentalists can play both saxophone and clarinet well.
But unlike the more cylindrical clarinet, saxophones have a very conical, flaring shape. Interestingly, the shape makes both blowing and fingering easier on saxophone than on clarinet. The simpler fingerings come from the fact that the conical bore causes the instrument to overblow - that is, to get its first usable overtone - at the octave, rather than at the twelfth (an octave plus a fifth). Notes that are one octave apart have essentially the same fingerings, and the saxophone does not need the extra keys that clarinets must have to produce the notes (from the octave to the twelfth) that would otherwise be missing.
Saxophones are usually made of brass (occasionally silver alloy or plastic), but they are still classified as woodwinds, not brass, because the sound is produced by a reed (not a cup mouthpiece) and the instrument is shaped and played like a woodwind, not a brass instrument. (Please see Wind Instruments: Some Basics for more on this.)
Of the four saxophones in common use, the soprano is the smallest and highest-sounding. It is straight and looks a bit like a metal version of a clarinet. The other three commonly-used saxophones all have an upturned bell at the end of the instrument. The alto is a bit longer and lower-sounding than the soprano; it is very popular as a solo jazz instrument. The tenor is a bit longer and lower-sounding than the alto, and the baritone (you may hear it called the "bari sax") is even larger and lower-sounding than the tenor. Tenor and bari sax, like alto, are both common jazz instruments, a standard part of a jazz "big band", for example. Soprano sax is a little rarer, but still not difficult to find.
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