The mouthpiece is one of the most important components of your saxophone. It always surprises me that some of the best saxes in the world are still shipped with cheap mouthpieces. Your mouthpiece is the root of all the tone of your saxophone and needs to be carefully matched to your own embouchure and reed. The reed that works for one person on their mouthpiece may be useless for you. You need to experiment with different hardnesses of reed and different tip openings to get a good tone. A wider tip opening (i.e. the gap between the tip of the reed and the end of the mouthpiece) is measured in all sorts of different ways by different manufacturers, which is summarised in the following tables. Generally, thousandths of inches or a 6* rating system is usual.
A wider tip opening will have a great effect on your tone. It is quite common for classical musicians to play closed mouthpieces while jazz musicians play open mouthpieces. Some of the more extreme designs are more like shark yawns than mouthpieces so always try a mouthpiece before buying it. You may be unable to get a note out of a mouthpiece if it's too open for you. Some experienced musicians, on the other hand, cannot play on a closed mouthpiece because their embouchure is too hard from years of playing. Choose wisely!
Remember that the ligature, which holds the reed, is also very important. There are some ligatures which are built on to the mouthpiece or hold the reed in several places. These generally have a different sound from double-clamping ligatures and may even claim to extend the life of your reed! If you want to experiment with a `soft-ligature', which will be a little more flexible in changing registers and altissimo (generally), try a shoelace. Seriously!
The material the mouthpiece is made from will make quite a difference. Commonly, ebonite is used. This is a hard rubber which will give you a fair sound but perhaps a little thin. There are notable exceptions where ebonite performs very nicely. It is the most common mouthpiece material, and the majority of players prefer it.
Personally I do not. Metal is also common and gives you a lot more power, as well as a completely different sound. Metal mouthpieces give your sax a fuller tone which can make for far more expression and better performance. Guess what I prefer. :) They are a little harder to play (especially on cold days - put them in your pocket first). Among jazz musicians, these are the most common choice. Other materials include plastics and wood, although these are more specialised. Generally, the tone you get depends on the density and resonant qualities of the material, as well as the size of the baffle and so forth. Try before you buy!
The following from Elias Haslanger refers to the Sugal Wood mouthpiece. I will add more such snippets as they become available.
"The mouthpiece is truly great--- for me. I cannot promise it will work for you but I immediately could tell a huge difference between the Otto Link I had been playing and my new Sugal wood. It will take a few months to get used to it but it is worth it. The tone plates do change the tone slightly but I like consistency so I chose the darkest and stuck with it. Also, the more you change the plates, the more you wear down the rails so I would avoid switching out the plates. If you get a wood piece you must oil it and take good care of it. I've had mine for about 4 years and it is starting to show some signs of wear. It wont last forever. :-(" - Elias----
"This is going to be a matter of personal taste. I'm lucky enough to be playing a Selmer Mark VI 120,000 series alto. I'm now on my sixth mouthpiece and the search for the best mouthpiece seems to be never-ending. I've tried the Selmer S80 C* (narrow tip opening allows saliva to produce an annoying gurgling sound after a while), Vandoren A27 (seems to take more air to blow than I'm comfortable with), Rousseau 4R (just not the sound I was looking for), Morgan 3C (a little too dark), and Selmer S80 D (if I try to use a stiff reed to get nice high notes my embouchure gets tired quickly).
My preference is for the Selmer S90 -180 with a Vandoren 2.5 or Peter Ponzol 3 reed - gives a nice sound in all registers without fatigue. Good Luck !"