Consider these general effects which apply to most brass mouthpiece dimensions and design:

Rim

Wide: Increases endurance.
Narrow: Improves flexibility.
Round: Improves comfort.
Sharp: Increases precision of attack.  

Cup

Large: Increases volume, control.
Small: Relieves fatigue, weakness.
Deep: Darkens tone, especially in low register.
Shallow: Brightens tone, improves response, especially in high register.  

Throat

Large: Increases blowing freedom, volume, tone; sharpens high register (largest sizes also sharpen low register).
Small: Increases resistance, endurance, brilliance; flattens high register.  

Backbore

Combinations of size and shape make the tone darker or brighter, raise or lower the pitch in one or more registers, increase or decrease volume. The backbore's effects depends in part also on the throat and cup.