Start by holding the artichoke firmly on its side and slicing the top 1/2-inch or so off with a knife.
Cut off the stem of the artichoke close to the base. Remove the smaller tough leaves around the base of the artichoke.
Cut off the tips of each leaf with scissors (it's not necessary, but makes a nice presentation and removes the spiky points).
Steam the Artichokes
Fill a pot large enough to hold the artichokes (they can be touching) about 2 inches deep with water. You can add a few ingredients to the water if you like (see Notes), but you might also just add a bit of salt and leave it at that.
Use a steaming basket or just put the artichokes into the water, cut side up or down (they can be squished a bit to fit). Cover the pot, bring the water to a simmer over high heat, then lower the heat and continue to simmer the artichokes for 25 to 40 minutes, or until a knife slides easily into the stem end or a leaf pulled from the artichoke comes out easily.
Serve the artichokes hot, warm, or cold, sitting on their stems with melted butter mixed with some lemon juice, mayonnaise (flavored or plain), or another dipping sauce, such as Lemon Butter Sauce.
Notes
You may want to have a cut lemon to rub against the top and tips of the leaves after cutting to prevent it from browning (this is not necessary, but is aesthetically appealing).
You can add some seasonings to the water for a light flavoring. These might include some wine or broth in place of some of the water, peppercorns, fresh or dried herbs such as thyme, rosemary, parsley, or bay leaves, and slices of lemon.
The steaming time really depends on the size of the artichoke, which can range in size from 3 to 6 inches (baby artichokes, which are even smaller, will steam in 15 to 20 minutes).