Starting at the top, a saxophone has the following major parts:
Mouthpiece assembly (mouthpiece, ligature, reed), neck assembly (neck, tightening screw), top joint and keywork assembly (top stack) , bottom joint and keywork assembly (bottom stack), bow joint (the U-bend at the bottom), and bell. On many if not most saxophones sized from alto to baritone, the top and bottom joints are actually one continuous conical tube.
The soprano and sopranino saxophones sometimes dispense with removable necks, and usually do not have bow joints (since most such saxophones are straight, like soprano clarinets). Contrabass saxophones may have additional joints due to double-wrapping of the tubing (e.g. the "Tubax" Eb and Bb contrabass saxophones).
Saxophone keywork is made of of many individual keycups, pads, resonators, rods, levers, springs, and touchpieces. I will not list these here. For more information, consult the literature on musical instrument repair.