For the player, width of cup correlates to the players jaw, front teeth and lip structure. Bigger teeth or fleshier lips can handle a wider cup like a Bach 3C to 1 1/4 C or Schilke15 to 18. Normal face structure, Bach 7C to 3C or Schilke 12 to 14.
The Bach rim is flatter, so if you want a rounder, more comfortable rim, the Schilke C,D or E would be better. The rounder rims do take a little more endurance though, as they allow the lips to slip a little, meaning you need a little more strength to keep the aperture closed.
The normal or stock throat or drill size should suffice. If you want more volume, you go a little bigger. Just beware, that once you start taking metal away, you can’t add it back.
Cup depth is a hot topic. The C cup is the standard among the major manufacturers. There can be 5 or more depth sizes. The A cups are the shallowest, Schilke/Yamaha version. Bach shallow would be D or E. Ease of upper register is better, but tone quality suffers in the normal range. The Schilke D or E cups are deeper and gives a darker quality and needs to have more air pushed through. The deeper cups tend to produce flatter pitches, while the shallow cups tend to push the pitches sharper.
Backbore taper can range from tighter to open. Here the analogy, of holding your thumb over the garden hose, making the water spurt farther, represents the tighter taper. So, a taper just smaller/tighter than normal or average should be fine for upper register work or if you want a little more resistance to push against. But then there are those who prefer less resistance and blow a fast volume or air in their playing.
These are all pretty generic thoughts, but to those who are just starting to explore the realm of brass mouthpieces…This will get you started. Of course, everyone plays a little differently, and I know that not all what I have typed, pertains to everybody. Always consult with your private instructor first. Have a great day!