1: There's no need to wash the asparagus before peeling - you can do that afterwards.
2:
Peeling
Place the asparagus on a cutting board, ideally a little higher than your work surface.
3: Using a peeler, make a movement from the head to the foot, stopping a good 1 cm from the foot of the asparagus, there's no need to go all the way...
4: ...then rotate the asparagus slightly and repeat, until you've gone all the way around the asparagus.
5: At the end, break off the stalk - chefs call it "the wood" - along with the peelings, and throw it away (in the compost).
6: Your asparagus is completely peeled, ready to be washed and cooked.
7: Here's a short video showing the complete peeling process.
8: Do this for all your asparagus, then wash and drain.
9:
Cooking "À l'anglaise"
This is the simplest method: bring a pan of water to the boil, season with salt, and plunge in the peeled asparagus. Cook until the asparagus melts, about 10 minutes. The tip of a knife should pierce the bottom of the asparagus easily ("as if in butter").
10: Then drain the asparagus...
11: ...then set aside on a kitchen towel or paper towel. This is important because asparagus cooked in this way always gives off a little water afterwards, which is best drained off.
12:
Cooking "À court mouillement"
A more sophisticated method, asparagus is cooked in a mixture of butter and water, which then reduces and concentrates the flavors. In a frying pan or saucepan over medium heat, melt a large knob of butter.
13: Pour in the peeled asparagus, season with salt and pepper, then stir a little, and add enough water to reach halfway up the asparagus (about 1 or 2 glasses).
16: Cook until melted, about 10 minutes, a knife tip should pierce the bottom of the asparagus stem very easily ("as if in butter"). Remove the paper and lid for the last 3-4 minutes to reduce the cooking juices.
17: Drain the asparagus, and it's ready for your recipe, or to be eaten as is, with a mayonnaise or hollandaise sauce, for example.
Remarks
Preparing green asparagus is a little different, see how here. For short-cooking, you can replace the water with stock, or a mixture of stock or water + white wine. Many recipes recommend cooking asparagus "in bunches", but I don't find this very effective: asparagus in bunches cooks less well than on its own.