Developed by Adolphe Sax as a woodwind instrument as previously mentioned, the saxophone is a term used for one of what is actually a single-reed instrument in a variety of sizes and ranges.
Once used in classical music, it's now mostly known as a wind instrument in jazz and early rock and roll, as well as reggae, ska and its various sub-genres including rocksteady.
Fundamentally, a saxophone is either a straight (in the case of most soprano saxes) or curved (in the case of about everything else) brass tube, sealed with leather pads triggered by spring-loaded keys. It follows the Boehms fingering as much as is possible so is somewhat easily picked up by players of the recorder, flute, and clarinet though each instrument has its own nuances in terms of how to cross the "break", the point where you go from the lower register of the instrument past the C represented by the third space in the treble clef and into the higher register.
Given that it IS a Boehms instrument, it's also a transposing instrument meaning that "C" played on the alto and baritone sax is concert pitched E flat, and tenors and soprano saxophones finger "C" but play B flat. It is therefore the responsibility of the composer to transpose parts from concert pitch or for the saxophonist to know which "keys" in his own instrument map to the concert pitch keys.
It used to be common practice to learn the clarinet first and then move over to the saxophone, but honestly given the differences in embouchure and the break it just makes more sense to start on the instrument these days.
The instrument has a surprising number of sound qualities, it's used in brash, cutting, sharp and bright ways in rock and roll to "cut" through a wall of guitar and bass with a gold-lacquered and brass mouthpiece, or dark and sultry with a hard rubber mouthpiece. Intriguingly, the mouthpiece is very important in terms of sound production and it being properly faced and matched to the player in terms of tip size and volume makes a giant difference. Vintage mouthpieces can fetch hundreds of dollars and any of a number of materials from rubber to crystal to brass have been purposed for this endeavor. The reed is made of cane and attached with any of a number of types of ligatures, taking over from the "tied on with thread" method which the instrument started with. The mouthpiece is slid onto the neck, which has a cork wrapper to seal the neck air-tight. The use of a Chap-stick style lubricant to avoid tearing or drying out of the cork is a necessity.
The reason for cane strengths is that a narrow tip requires a tighter embouchure and a harder reed, a larger amount of tip space can be matched with a softer reed to allow for wider dynamics.
Many saxophonists use a reed tool to face, edit, or otherwise customize the shoulder or thickness of the reed to make it ultra-playable. Synthetic reeds are now available which do not need to be moistened before use.
The instrument is usually slung around the neck with a strap that attaches to a loop brazed on to the back of the saxophone, and adjusted so the player can stand in a neutral position with the mouthpiece at mouth level. Heavier instruments like the tenor or baritone are sometimes attached to a chest harness to take stress off the neck, and contra-bass saxes are typically mounted on a stand.
To play the saxophone the lower lip is drawn over the teeth and placed on the reed, mounted on the underside of the mouthpiece. The mouth seals around the rest of the mouthpiece proper and tension is placed on the reed with the lower lip while air is blown past it. It is an easier embouchure to learn than the clarinet or trumpet but its execution affects everything from being in tune to the kind of tone to volume, so needs to be learned properly from a qualified teacher as quickly as possible. The note played is a function of the octave key (or overtone production, similar to a trumpet) and the keys used on the instrument to make it "longer" or "shorter" before air can escape, similar to a flute.
Care must be taken with this instrument. The keywork and the collection of blued steel springs and mechanisms are complex, delicate and easily nudged out of alignment or will shrink and leak. By definition the leather used in the pads will eventually degrade or cork used in the neck will shrink or tear, and leaks are the bane of the musician in some instances making the instrument unplayable. "Re-padding" an instrument involves its partial disassembly, the melting of the resin used to attach the pads, and new pads being added. Sometimes "re-floating" a pad is done, namely the careful use of a propane or butane torch to heat the resin to melting point, and then allowing it to solidify with the key pressed down, so that the pad adjusts and seals. Because this is expensive and error-prone (the decorative lacquer can burn) saxophonists are advised to disassemble and "swab" the instrument with a weighted cotton wicking assembly after every use. The instrument is also to be taken care of as being jostled, bumped, dropped or put in an ill-fitting case can damage or bend important structures.
The sax has come a long way. Modern saxes have more keys (the F# key under the D key is an example), better springs, and many now allow micro-adjustment with a tiny screw to avoid the need to "float" pads as often. But you will still find sax men carrying a small tool kit to do emergency field repairs.
For an idea of what the various sizes and registers are used for: Kenny G is the best known soprano saxophonist, putting out slightly out of tune, tinny schmaltzy songs. Should also mention Sidney Bechet, who not only was a soprano player but was instrumental in putting that instrument in jazz in the first place. Paul Desmond of the Dave Brubeck Quartet and Charles Parker are some of the best known alto saxophonists, whereas John Coltrane and most rock saxophonists play the tenor or even the baritone. The baritone is famously used to counterpoint many early rock and roll songs as well as the theme from The Muppet Show.
At or around the 1920s it was in vogue to have "C melody" versions of the alto and tenor meaning that you could play along with a pianist directly from their sheet music, but these never caught on and are mostly found in vintage stores and pawn shops.
Saxophones are almost always brass, although some have been made out of composite materials or even, disgustingly, cheap plastic for the learner. Avoid plastic saxophones. Please. Avoid. The finish on the other hand tends from the common-in-use golden lacquer, to silver or black nickel finishes. For a while in the 1980s it was in vogue to have colored finishes, but that's typically seen as gimmicky and only Cannonball produces a "Ruby" model these days, with the exception of many of a number of "starter" instruments on Amazon only worth hanging on the wall.
That being said, most saxes are from Taiwan these days. There's a certain amount of angst at the French finally giving up and Selmer, the "Gibson" of saxes having made many legendary models that command high prices on the used market - being acquired as a brand by a Chinese manufacturer. The ones that are forged there and hand-finished in the US remain usable, in fact useful instruments.
It remains an instrument in demand because unlike the piano or drums it is a tricky instrument to "sample" and play on a synthesizer. It comes out sounding like the "look at the sexy woman walk by" riff played poorly on a sampling keyboard, with obvious mod wheel bend, in a cut-rate Chinese kung fu comedy. But that hasn't stopped rap and hiphop from sampling heavily from the Blue Note catalog or grabbing sax squeals or growls as rhythmic "stab"s from classic recordings, and you can still hear the occasional blast from the past in modern music.
Many people who want to play the sax think a $300 model on Amazon is a good idea. Either that or the $400 used model in the pawn shop sold "as is". Problem there is, they're usually more trouble than they're worth. Shoddy materials, poor workmanship or the vicissitudes of age have rendered most of these instruments tricky or near impossible to play. Buy a reconditioned one from a reputable dealer if you must, but prepare to eat some cost. It is, however, totally worth it.