A good mouthpiece can make a fair horn play like a good horn...it is VERY important. I recommend beginners start with a medium, middle of the road mouthpiece. New instruments by Yamaha and Selmer tend to come with very good beginner mouthpieces. There are so many options and new brands being manufactured all the time. Mouthpieces can vary widely among the same model due to manufacturing inconsistencies. In addition, the numerical systems used to describe the size and type of mouthpiece differ widely from maker to maker. One brand's large is another brands small. Don't go by size alone. My #7 mouthpiece is not the same as someone else's. As a very general guide for beginner mouthpieces, Hite, Yamaha, and Selmer offer good options. The real proof of a good mouthpiece - does it make the instrument easier to play? Does it make YOUR sound easier to obtain? Does it play as easily on the low notes as on the high notes? Does it play in tune? The answer to these questions should be yes.
Mouthpieces are manufactured from a variety of materials; most often plastic, hard rubber, or various alloys of metal. Exotic mouthpieces made of materials like glass, hard wood, copper and other metals can also be found. I've found that the internal shape of the mouthpiece, or baffle, has more impact on the tone than the material, although a material like metal will tend to sound more edgy than hard rubber. A high baffle, where the inside roof of the mouthpiece comes down close the reed, or is more shallow, will have a louder more focused tone than a mouthpiece with a deeper baffle, where the roof of the mouthpiece is deeper and not as close to the reed. The same is true for the chambers; a smaller chamber or bore will produce a more penetrating sound than a larger bore. The power and projection you get from the mouthpiece come at a price though... The lower notes on the instrument become less stable and harder to play; and after a certain point the intonation or tuning of the instrument begins to suffer. In addition to the baffle and the chamber are the tip opening and facing. The tip opening is the measurement from the tip of the mouthpiece to the tip of the reed when you are looking at the mouthpiece head on. A smaller tip opening is easier to control, but a larger tip opening allows for a larger volume of air and more room for scoops, bends and effects by the artist. Generally the smaller the tip opening the harder the reed used. Orchestral or 'classical' saxophonists favor smaller tip openings. Jazz saxophonists tend toward larger tip openings. The facing is the measurement of the tip of the mouthpiece to where the reed and mouthpiece meet together viewed from the side. A short facing lends itself to a strong upper register and fast response at the expense of the low end. A long facing is exactly opposite, less fast to respond and making the low end easier and lusher but the high end more difficult. In between is the medium facing and this is what I play.
I prefer to have one mouthpiece that allows me to do a little of everything. I use an older Meyer currently (don't know the specs, medium chamber, medium open) on alto and an old Hollywood Dukoff #7 on tenor. The sound is in your head and no matter what mouthpiece you play on, you will tend to revert back to that natural tendency (until that concept changes). So, pick a mouthpiece that lets you make your sound with the LEAST amount of EFFORT.