Consumer Info on:
Materials - Hard rubber, metal, plastic, crystal or wood. The vast majority of desirable are made of hard rubber. Crystal mouthpieces are highly respected, but the risk of shattering is always there. Metal mouthpieces are exclusively for saxophones. Most of the cheaper mouthpieces are made of plastic. A particularly good plastic model is Hite Premier. Wooden mouthpieces are fairly rare. I don't recommend these for students because of the amount of saliva that mouthpieces have to endure. I feel that maintenance issues make wooden mouthpieces impractical.
Sizes - Mouthpiece sizes are mostly expressed in terms of the tip opening. This is the distance from the tip of the mouthpiece to the tip of the reed. This distance is measured either in millimeters or in thousandths of an inch. A second measurement that may be of interest is the facing length. This is the distance from the tip of the mouthpiece to the point where the reed makes contact with the rails/table.
Sizes vs. Playing characteristics.
Tip Opening - closer facings tend to have more resistance, greater stability and work better with harder reed strengths. More open facings tend to be freer blowing, more flexible and involve softer reeds.
Facing Length - Shorter facing lengths tend to optimize performance in the instrument's upper range. Longer facing lengths tend to favor performance in an instrument's lower range.
Brands: Vandoren and Selmer are the most common brands. Vandoren's B45 is the world's best selling model, and its 5RV is a close runner up. Selmer's HS* is the model included in the case with most of its best clarinets. It's a closer opening, while the HS** is a more open tip. Hite mouthpieces are fairly common now, and their Premier model is an excellent first mouthpiece for a student. Gigliotti's are excellent mouthpices and many sax doublers like the Portnoy.
Sizes: Most American classical players prefer closer tip openings and harder reeds. They provide excellent stability and pitch. Many jazz players and sax doublers prefer medium or open facings with medium or soft reeds.