Covent Garden Market is mainly for tourists, but I still loved going once in a while. When we went to the center of London, sometimes it felt like the center of the world!
Covent Garden Piazza has a multitude of shops as well as a bunch of places to eat (including a Ponti's in the center with reasonably priced sandwiches and pasta). It's really a hub for nightlife and culture and is definitely worth a visit.
Venture further into the Piazza and you'll see the covered market with arts and crafts as well as touristy items. I believe there are also some food stalls upstairs.
The best part about Covent Garden is the people watching—and the free entertainment. On a given day, there may be musicians playing in the center of the Piazza or a crowd gathered around a Punch & Judy show. You may see a magician or a juggler or a guy who rides a unicycle.
Covent Garden is also great for kids—as long as you make sure you know where they are at all times!