Maltby Street Market (Sat-Sun, 9- & 11-4) is a great under-the-radar one to visit, in an atmospheric former ropewalk (now made of bunting). It's not huge, but it's a good local haunt for a late weekend breakfast - boozy but delicious if you go for the waffle stand and Little Bird Gin's stall. Some interesting vintage goods and antique stores under the arches, gourmet and baked goods, and cheeses. Go for: the chill vibe, the gin.
Borough Market (10-5 except Sunday) is always impressive and lovely. It's the one everyone knows, so everyone goes, but that doesn't make it any less worth visiting. It's often crowded, but the array of produce and prepared foods can't be beat. Pop into Southwark Cathedral just next door for a complete visit. Go for: everything! Do a lap to take it all in, grab the freshest, cheapest produce you can find, and if you're there for breakfast the Bread Ahead doughnuts come highly recommended.
Real Food Market (Fri-Su 12-6) is a good lunchtime pick with an array of internationally-inclined food stalls. Just behind the Southbank Centre. Go for: lunch.
Marylebone does a nice Sunday market (10-2) with a good array of fruit, veg and meats; accordingly a bit pricier because of the area, but a nice, established (...establishment) family vibe. Go for: a posh organic shop.
Bloomsbury Market, by contrast, is the student scene, and on Thursdays (9-2) is perfect for a lunch break visit. It has some produce, but mostly baked and prepared foods. Certain stalls (lamb and chicken wraps, pasta and paella) are worth braving the lunchtime lines of UCL students. If you don't mind lines and really don't like paying for things, you can wander beyond the market into campus where the Hare Krishnas give out free Indian food at lunchtime. Go for: a meal, the bio apples/juice stall.
Piccadilly Market (M-Sat, 10-6ish) is just outside a Wren church (St. James'), with mostly antiques and crafts stalls, and even a few fossils. It's pretty and interesting to browse, and has a nice green square to sit in. Go for: Hatchard's bookstore nearby, then pop in to the market and church to escape the streets of Piccadilly.
Portobello Road (9-6 except Thurs) is well known for shops and antiques, and you should have a gander at nearby Golbourne Road as well. There's a really friendly record stall, lots of cheap home goods and vintage bric a brac. Go for: the experience.
Brick Lane, Spitalfields, and Petticoat Lane are all fairly near each other, and Sunday is the best/biggest day to go if you want to see everything in action. Brick Lane is best for Indian and ethnic foods, Spitalfields (Th-Su, 10-5) and Petticoat Lane (M-F smaller, Su 9-2) provide vintage, antiques, art, fashion, and almost anything you can think of. More down and dirty, crowded and crazy than Portobello. Go for: International food, fun.
Camden Market (10-6 every day) has excellent street food stalls along the lock/river; there are nice crafts in the market hall, and a lot of cheap tourist goods along the high street. Camden is one of the most crowded markets at the weekend, especially around the food, but it's worth it to see the crazy crush of people from all over - international tourists, quirky locals, and visitors of all different social statuses. Go for: the people-watching, food stalls, cheap clothes.
The major markets that I have not yet been to, but hear delightful things about, are Columbia Road Flower Market (Sun 8-3) and Broadway Market (Sat 9-5). Will update as I can!