There are two kinds of Asparagus Peelers. Both are made of stainless-steel, and almost all makes will be dishwasher safe.
The most common kind of Asparagus Peeler looks like a set of tongs. The arms are generally 7 inches (18 cm) long, and about 1 1/2 inches (4 cm) wide. One arm has a few wavy curves in it at the end; the other arm has a vegetable peeler blade part on the inside of it. You press the arms closed upon a stalk of asparagus, and pull the stalk through them. The bottom curvy arm helps to hold the asparagus in place and guide it. Then you rotate the asparagus stalk to expose another side to the blade, and pull it through again, and repeat until the asparagus stalk is all peeled. Some makes have a fastener to keep the two arms together for less-awkward storage in drawers.
Another type of Asparagus Peeler is just a single handle peeler. On the metal shaft part, there's a separate thin metal blade (you may see this referred to more dauntingly as an "flexible articulated blade") that you raise from the rest of the metal shaft, which helps to guide the asparagus stalk as you pull it through between that raised thin metal blade, and the metal shaft itself. The edge of the thin metal blade does the actual peeling.
You can also use Asparagus Peelers for peeling things such as older runner beans, etc.
Green asparagus doesn't usually need to be peeled, though some people prefer to peel the thicker ends of them. White asparagus eaten raw doesn't need to be peeled, but cooked white asparagus does always need to be peeled, as its skin goes tough when cooked.
You can use the asparagus peelings to make a broth or soup.