Artichokes are easier than you might think.
Step 1: Rinse and Trim
Find some good-looking artichokes. I usually get a bunch of the small cheap ones instead of the big expensive ones, because the leaves are usually younger and more delicate. Brown on the outer leaves is fine- it's caused by frost, which can help concentrate the flavor. Mold, and black or mushy spots, are to be avoided.
Wash and drain your artichokes, then chop the top layer of prickly spiny bits off. While trimming isn't absolutely necessary, the lack of spines will be appreciated when you peel the artichoke later. It can also help facilitate more even steaming.
Especially with the larger, coarser artichokes, some people prefer to use scissors to trim off the spike from lower leaves as well. I see this as another good argument for buying the smaller, younger, cheaper artichokes which rarely have spikes below the top leaves.
Step 2: Steam
Bring a pot of water to a boil, and drop your artichokes into the steamer section. You can add half a lemon or lime to the water to preserve a bit of the artichokes' color.
The cook time will depend largely on artichoke size; you'll just have to guess and check. Check by stabbing through the thickest part of the base with a paring knife- when the knife goes through without resistance, the artichoke is done. These little guys took just under 20 minutes.
Boiling is also acceptable, but leaches more vitamins and minerals from the artichokes and leaves them a bit soggy and harder to handle. If you've got a steamer, use it.
Step 3: Make Sauce
While the artichokes are steaming, make some sauce.
Melted butter is traditional, sometimes with lemon and/or garlic.
I prefer something a lighter lemon vinaigrette, which looks/tastes much like lemon/garlic butter:
1 large lemon, juiced
extra virgin olive oil, equal to lemon juice volume
fresh grated garlic (I use lots; do this to your taste)
dab fresh mustard as an emulsifier
dash salt/pepper to taste
Whisk the ingredients together; you'll end up with a gorgeous buttery-looking emulsion. Drop a few tablespoons into individual ramekins to avoid cross-contamination.
Step 4: Remove Outer Leaves
Peel off and discard the small lower and outer leaves. These don't have much flesh at the base, and are a bit tough. They're standing between you and the good stuff, so out they go.
Peel down the stem as you go; this will come in handy when you reach the heart.
Step 5: Eat Large Leaves
When you reach the "hips" of the artichoke, you'll begin pulling off leaves with a large white attachment point at the base, and some surrounding fleshy tissue. From this point on, all of the leaves will be at least partially edible.
Remove the leaf, dip it in your sauce, and put it in your mouth upside down so the inside bottom part sits against your lower teeth. Drag your teeth along the leaf, scraping off the fleshy tissue down to the attachment point. Discard the rest of the leaf, as it's too tough to eat.
Step 6: Eat Center Leaves
The softer center leaves can be removed in one piece, and the lower 1/3-1/2 of the leaves are good to eat.
Wiggle the cap of center leaves; they should release easily, coming off together. Dip the entire base in the sauce, and eat from the bottom up. The leaves will thicken part way up, making it fairly obvious where to stop eating.
Step 7: Remove Choke
The artichoke is a relative of the thistle, and if allowed to mature they open to display a lovely purple flower. The cluster of immature florets at the center of the artichoke is known as the "choke", and should not be eaten.
Remove and discard the last small leaves covering the choke; they usually pick up some of the fuzzy bits, and so are best simply disposed of. Use a spoon or your finger to scrape out the soft, furry choke without losing any of the tasty heart below.
Step 8: Eat Heart
Dip the remaining heart into the sauce, and eat. Your own ramekin of sauce is advised, as you'll be double and triple-dipping. Eat the heart, and chew down the stem until it starts to get stringy. Good stuff.
Step 9: Extra Bits
Keep a couple of bowls on the table to discard leaves, chokes, and stems. Plates full of artichoke detritus aren't fun, and don't leave room for the rest of dinner.
Artichokes keep well in the refrigerator, and are great served cold the next day. Just make sure you've got more tasty sauce to serve with them.