I LOVE pepper. All kinds. On my spice shelve, about half the jars contain some form of pepper. Black, white, green, pink, mixed, mignonette, szechuan, tellicherry, etc...
Black is definitely the go-to pepper. It has that "classic" black pepper taste, at least to me, since I grew up with it. It's produced from unripe berries from the pepper plant.
FUN FACT #1: Peppercorns are fruits. So next time someone wants to show off and convince you that "tomatoes are not a vegetable, but a fruit" (that statement is wrong BTW), you can tell them that peppercorns are fruits too!
Green peppercorns are immature black peppercorns. They're not as hot as black pepper. It's less spicy but has a bit more flavor than black IMO. It works really well on duck, fish, chicken... but also on steak! I actually use both the dried green peppercorns for rubbing on meat or fish, and the canned green peppercorns in sauces, they are quite different.
Pink pepper is not pepper (despite its name and appearance), but it wasn't died pink for appearance's sake, no. In fact here in Los Angeles you find pink pepper trees at every street corner. It's got a very subtle lemony/anis kinda flavor which is just perfect with fish and poultry, but just really really good with fish.
White pepper is black pepper without the skin. It is classically used instead of black pepper in white dishes (like a velouté or a béchamel for example), although it does have a different flavor, and personally I really don't mind the black specks in my white food. But white pepper is good too - kinda hard to describe the difference.
Mignonette is just black and white pepper mixed together.