The most basic way to enjoy an artichoke is to steam it. You steam them whole after cleaning them carefully. When you eat the artichoke you start by peeling off the leaves and scraping off the flesh inside the leaf with your teeth. You can enjoy the leaves with mayonnaise, melted butter, lemon juice or even different types of vinaigrette. After eating the leaves you will have uncovered the heart, which is covered in the fuzz, which you gently scrape off with a spoon and discard. Then you can enjoy the heart.
