The mouthpiece, not the saxophone, is responsible for over 80% of the tone a musician gets while playing. And more than half of the sound that comes from the mouthpiece comes from the baffle. The closer you get to the initial source, or creation point, of the sound wave, the larger the impact you have on the sound. Since the baffle is the very first thing that the sound wave hits, it is where the initial shape of the sound is created and has the largest impact on the resultant sound coming out of the saxophone/mouthpiece set-up.
This is why the baffle is the single most important part of the mouthpiece/horn configuration. For reference, the baffle is the section of the mouthpiece directly behind the tip rail that extends back into the mouthpiece about a half inch. While a good facing makes the mouthpiece easy to play and respond, the baffle is what creates the basic sound of your horn. The baffle is where the sound is first created. So while classical, rock, and jazz players may play the same horn, and even have the same facing and chamber size, where the basic sound difference comes from is the baffle. There are four fundamental baffle shapes that a mouthpiece can have: Straight, Roll-Over (Convex), Concave, and Step.
The Straight baffle has a straight line from the tip rail into the chamber. It has very dark sound. The first saxophone mouthpieces by Adolf Sax at the turn of the century until the early 1940's were all straight baffles. This baffle can help keep musicians who blow very hard from sounding too bright.
The Roll-over baffle forms a convex curve right behind the tip rail which acts like a very short rounded high baffle. This gives the mouthpiece added edge and growl while also allowing a warm tone. This is the classic sound of Otto Links and Meyers from the 1950s to present. A good roll-over baffle is an art form to create.
The Concave baffle looks like an indentation behind the tip rail. This is seldom used as it produces a dull tone unless combined with a step-baffle.
The Step baffle forms a high step dropping down into the chamber. It creates a bright sound and great projection, so is often used in R&B, Rock&Roll, and Smooth Jazz. This baffle can sound thin if not done properly.