Though they are made of brass (or other metals), saxophones are considered woodwind instruments. This is because of the way they sound and how they're played. A saxophone works in the same basic way that any horn-shaped instrument does: Something in the mouthpiece area vibrates and the sound is amplified by the resonance of the hollowed instrument. (It gets a bit more complicated than that, of course, but here's where we'll start.)
Reed and Mouthpiece
The mouthpiece of a saxophone consists of a sort of hollowed cylinder shape--"sort of" because it tapers on one side into a narrow edge, an edge that is open on one side. This is where the reed goes. The reed is a thin, soft piece of bamboo cane that makes the heart of the saxophone sound.
Have you ever held a blade of grass tightly between your hands and blown on it to make it whistle? The reed on a saxophone works in essentially the same way. It's attached tightly to the mouthpiece of the saxophone (using an adjustable metal fastener called a ligature), then the player holds the mouthpiece and reed between his lips and blows air with enough pressure and velocity to make the reed vibrate. This vibration will resonate throughout the oral cavity of the player and also throughout the instrument.
The pitch of the notes being played is partially determined by the player's use of air and mouth pressure, but the biggest changes in pitch happen through the covering and uncovering of holes. When more or fewer holes are covered in different places on the instrument, it changes how the instrument resonates; the more holes covered, the lower the pitch, and the fewer holes, the higher. In this way, a saxophone works in the same way as a basic flute or whistle with holes that are covered by the player's fingers. However, the saxophone has so many holes, and many that are so large that a player cannot cover them all with fingers. Instead, the fingers operate a series of easy-to-reach buttons that operate mechanisms. These mechanisms cover the holes and are equipped with soft pads that will create the same air-tight seal that human skin would. Different combinations of these buttons allow different combinations of holes to be covered, and in this way, the saxophone can play a wide, chromatic range.