Your mouthpiece will have a big impact on your playing (sometimes even more than your instrument) and it is important that you choose the right one. When choosing a mouthpiece, you will have to consider a few factors like the size of your lips, your embouchure position and what you want to do as a player. Here are a few guidelines but keep in mind that it can vary a lot from one person to another.
-In general, a deeper cup will help your low register and a shallower one should improve your high range. You'll have to find a compromise if you want to sound good in both registers.
-Choose the diameter of your rim according to the size of your lips.
-Larger and deeper cups tend to give a darker tone most of the time as a "cone" shaped mouthpiece will produce a brighter sound depending also on the player of course.
-A larger bore should help you playing louder but might make it harder to play soft.
The best thing to do is to go to a music store and try out all their mouthpieces and pick the one that feels and sounds the best. If they're really nice, they'll sometimes let you take a few mouthpieces home to try. If you're in school, find the "mouthpiece box" and fool around a bit. Ask your friends if you can try their mouthpiece also. Don't forget to wash it before and after you try it!
Pay attention to how your chops feel when you try a mouthpiece. I find that the better it feels, the better it sounds.
Keep in mind that although a different mouthpiece may solve a certain problem, it may weaken another aspect of your playing so you'll have to be careful about that. It can also take a while to get used to a new mouthpiece but if after a couple of weeks, it still doesn't feel and sound right, you might want to consider searching again. Be patient but don't be stubborn. I played a certain mouthpiece for a couple of months that sounded very good but was hard to play in the high register and made me very flat. After a while, I was back to the shop to choose another one which I still play after 20 years!
All these guidelines are very subjective and there is no absolute truth about mouthpieces. Everyone has different muscles, teeth and lips so you'll have to proceed buy trials and errors to find the right one for you.
I am associate principal horn of the Singapore Symphony Orchestra and the author of the progressive methods. I'm happy to share my experience as a horn player and teacher with you.