The first step is to cut off the top. Cut off enough to reach the pinkish/purple portion in side.
The next step is to trim the prickly ends of the leaves off. The best tool for this task is a kitchen shears or scissors. Oh, and this is important. See that bowl in the upper right hand corner of the following picture? That bowl has some acidified water in it. Artichokes, like so many other types of produce, oxidize and turn brown when exposed to air. Be sure to have this bowl ready to dip the artichoke in periodically to prevent browning. You can acidify the water with lemon juice, vinegar, or citric acid.
While some recipes require the artichokes to remain whole, I prefer to cut them in half. I think it makes them steam more evenly. It's also much easier to remove the choke if they are in half. If your recipe needs them to remain whole, skip this step.
Now pare the tough outer layer of the stem off with a paring knife. This step may involve removing some of the lower leaves to reach the tender inner stem. Notice that I am keeping the spare half in the acidified water.
Artichokes brown very quickly. It doesn't look very attractive all browned, does it? So be sure to have the water ready and dip frequently.
Here's the choke or the "thistly" part that needs to be removed. As you can see, it wouldn't be good to eat.
Use a large spoon to scoop the choke out, scraping to clean it out thoroughly. If you are keeping your artichoke whole, this can be tricky. Peel back the leaves enough to access the center and scoop out. You may need to cut the tops back more when preparing them whole than when you are halving them.
And that's it! Now they are ready to use in your favorite recipe.