The artichokes discussed in this node are sometimes called globe artichokes to distinguish them from Jerusalem artichokes; the two are not related, though some say they taste similar. I'm going to discuss the preparation of globe artichokes.
Whether or not you trim your artichoke and how long you cook it will depend on its size. Small or baby artichokes are about the size of a chicken egg, while medium ones are about the size of an orange; any bigger than that and the artichokes are likely to be tough and dry: don't bother with those. Whether buying small or medium artichokes, rub the leaves together to check freshness; they should squeak, indicating that the leaves still retain moisture. Try pulling on a leaf; it should snap off, rather than bend.
No matter how you prepare your artichokes, note that they will begin to brown as soon as they are cut, so if any cutting is to happen, it's necessary to submerge them in water containing some acid to neutralize the enzymes responsible for browning. Some recommend vinegar, but I prefer lemon juice because it has a nicer flavour. Just plunge them into a large bowl of water containing the juice of half a lemon. Use the juiced lemon half to rub any exposed parts as soon as you've made the cut. Note too that artichokes will discolour expensive carbon steel knives, so be sure you use stainless steel when making the cut.
Okay, preparation. Smaller artichokes need less preparation than medium ones because you don't need to remove the choke, or small hairy bit on top of the heart. In fact, you can roast small artichokes, a wonderful way to concentrate their flavour. First, cut off the top quarter of the artichoke and snap off the outer layer or two of leaves, till you reach the yellow leaves. Remember to rub the cut leaves with lemon juice. Trim off the dark green exterior from the bottom of the artichoke and from the outside of the stem (but leave the stem, it's tasty), then carefully cut in half lengthwise. Drop the prepared halves into the lemon water as you work. When they're all done, drain them well and toss with olive oil, season with salt and pepper, and place them cut-side down on a cookie sheet (lined with parchment paper if you've got it, to make cleanup easier) and bake at 400°F (210°C) for about 25 minutes, turning after 15 minutes. They are done when you can easily pierce the heart with a skewer. While they're still hot, sprinkle with a little more salt and some lemon juice and serve, about four halves per diner. Just cut 'em up and eat 'em; they're great!
Medium artichokes can be served whole, but I do recommend trimming, because the ends of the leaves have sharp thorns at the tips that can prick diners; these are thistles, after all. Hold the artichoke by the stem and use scissors or shears to trim off the tips of the leaves. Cut off the top quarter of the artichoke with a sharp knife, then cut off the stem so the artichoke will sit flat. Drop each artichoke into lemon water as it's prepared. When you're ready to cook, I recommend that you steam them for the fullest flavour. Steam over water for about 30 minutes or until an outer leaf releases easily when pulled. Let them sit for a bit before serving, and be careful: they retain heat, and it's easy to burn your fingers. (You could chill them and serve them cold, too.) You know, of course, to have a look at how to eat an artichoke for tips on how to consume one with style and panache. One or two per diner should do.
Suppose you're a real sucker for punishment, and you want to extract the artichoke hearts and cook them in some cunning way? Use medium artichokes for more yield. Bend back and snap off the thick outer leaves, leaving the edible bottom portion attached, until you reach the light yellow cone at the centre of the thistle. Use a paring knife to trim off the dark outer layer at the bottom - the base of the leaves you've snapped off. Cut off the purple tip from the yellow cone. Peel the stem and cut off the bottom bit, but leave the stem attached. Cut the artichoke in half lengthwise and scrape out the small purplish leaves and the fuzzy choke from the centre. Drop each half into lemon water as you're finished. You could steam, braise, saute, or grill these babies, or slice them thin and use them in pasta sauce. This is a lot more work than using canned artichoke hearts, admittedly, but also a lot more flavourful and impressive.
Another word about artichokes: they contain a unique acid called cynarin which, for some people, stimulates the sweetness receptors in the mouth and makes anything consumed immediately afterward taste sweet. So for some people, artichokes and wine is a bad combination: your lovely Chardonnay will become Welch's grape juice in their mouth. This is genetically determined, and only experience will reveal if you or your guests have this quirk. sneff informs me that if the artichokes are coated - say with polenta - and then fried, it counteracts the effect of cynarin, though he doesn't know why. While I can't imagine coating and frying my artichokes, he's says it's a popular preparation in the south of Italy.