The most basic parts of a sax mouthpiece are the tip opening, inside chamber design, including the baffle, and the materials the mouthpiece is made of. The design features then interact with the sax reed and ligature to create the quality of sound for all the various notes you can play on a saxophone.
Student saxophones commonly include a basic plastic mouthpiece, which is designed for a new student. The tip opening will be pretty small or closed, the inside chamber more open and free, and the end product will work well with a medium-soft reed and basic ligature.
Mouthpiece makers have been designing basic saxophone mouthpieces for years but have also been developing options for different players and different styles of music. Every part of a basic mouthpiece has been experimented with and many of those options are available in the marketplace today.
You can find really loud and brighter, soft and fluffy, dark classical, old school jazz, rock and pop, classic R&B, and middle-of-the-road, all-around mouthpieces, to mention just a few.
Often, loud and bright go together because high baffles with smaller chambers create forced air and a brighter tone. The baffle area is the inside floor of the mouthpiece looking from the reed side, or the inside top of the mouthpiece. A high baffle will reduce the space between the reed and mouthpiece and greatly increase volume when you blow hard. The trade-off is a brighter tone.
If a player needs to cut through in a loud jazz or rock setting, these mouthpieces really rock. The classic Guardala saxophone mouthpiece sound is bright and edgy on tenor. The higher the baffle, the louder and brighter they seem to get. These are also plated metal mouthpieces and I believe the silver is the brighter of the plating options.
On the other far end of the spectrum are the classical, dark, and softer volume pieces like the Vandoren Optimum sax mouthpiece or Selmer S80 hard rubber mouthpieces. These have very open chambers, low baffles and come in smaller tip openings for harder reed use, and they produce a dark rich sound.
Hard rubber is a great material for saxophone mouthpieces and is much more durable than plastic. The sound combination of the material, low baffle, open chamber, and medium tip opening makes these a great upgrade mouthpiece for younger students and classical saxophone players looking to round the sound out a bit.
The middle of the road would be mouthpieces one could play in big band or in concert band. The Vandoren V16 saxophone mouthpiece comes in varies chamber sizes for more or less brightness from a good solid design. The classic alto sax Meyer mouthpiece sound is full but still projects when you lean into it and blow. Jody Jazz has a saxophone mouthpiece designed like the old Meyer called the HR* for the alto and the old link rubber design is called an HR* for tenor sax.
The basic design of a sax mouthpiece is always the same. The modifications are designed to give you more of something. Often, the balance shifts one way or the other because you can't get more of one thing without sacrificing something else.
A really loud and edgy Claude Lakey alto sax mouthpiece might be great for playing lead alto in a big band but be very difficult to play a nice ballad on. There are times that you need to play quiet or darker too. Many sax players have a variety of mouthpieces they use in different settings. I have always tended to look for one great all-around mouthpiece in hard rubber and a great jazz, pop, rock medal mouthpiece for the electronic settings.
It doesn't really matter how you decide to cover the issue, but understanding how mouthpiece design affects saxophone sound, will make for much more informed decisions when shopping for the next step toward that perfect sound on your sax.