London's top markets are (in alphabetical order):

Who: Alfies Antiques Market
What: London's largest antique market is located in an historical street market in Church Street, Marylebone, minutes from the West End. The range and quantity of stock is breathtaking and includes Art Deco, silver, furniture, paintings and prints, jewellery, ceramics, vintage clothing and much more. Alfies is the also the store of choice for celebrities such as Brad Pitt, Kylie Minogue and Hugh Grant.
Where: 13-25 Church Street, Marylebone, London NW8 8DT
Nearest tube / rail: Marylebone

Who: Bermondsey Market
What: an antiques market just south of the river, famous for its ill-gotten loot trade. Now one of London's main antiques markets, it attracts dealers and serious collectors. It is open Friday from 04:00 - 15:00, but many stalls start closing around noon. If you want to pick up a bargain, get there very early.
Where: Long Lane and Bermondsey Street, London SE1
Nearest tube / rail: London Bridge, Borough or Elephant & Castle

Who: Billingsgate Fish Market
What: the UK's largest fish market, with 54 traders, it is open Tuesday to Saturday from 05:00 - 08:30, with a much smaller shellfish only market on Sunday from 06:00 - 08:00
Where: Trafalgar Way, E14
Nearest tube / rail: Canary Wharf

Who: Borough Market
What: the oldest market in London, which still retains a Dickensian air. Go on Friday afternoon or on Saturday for the fine food market.
Where: 8 Southwark Street, SE1
Nearest tube / rail: London Bridge

Who: Brixton Market
What: Brixton Market if famous for its fruit and veg, second hand clothes and African food. This is a colourful bustling market.
Where: Electric Avenue
Nearest tube / rail: Brixton British Rail Station

Who: Camden Market
What: Camden market is really a collection of 6 individual markets, all close together so that they spill into each other. Probably best known for its 'alternative' fashion stalls, many of the markets are open 7 days a week. The market is at its busiest on Saturday and Sunday, when the 'Canal market' adds to the attraction. Great for a weekend stroll and for teenagers who want to shock their parents.
Where: Camden
Nearest tube / rail: Camden Town or Chalk Farm

Who: Chapel Market
What: a fruit, veg and clothes market in the heart of trendy Islington.
Where: Angel, Islington
Nearest tube / rail: Angel

Who: Columbia Road Market
What: a Sunday morning flower market in Shoreditch. Market gardeners from in and around london sell flowers, plans and anything garden.
Where: Columbia Road, between Gosset Street and the Royal Oak public house, London E2
Nearest tube / rail: Old Street or Bethnal Green

Who: Greenwich Market
What: Unique and distinctive arts and crafts, known for designer makers and small cottage industry.
Where: 11A Greenwich Market, off College Approach, SE10 9HZ
Nearest tube / rail: Greenwich or Maize Hill DLR

Who: Petticoat Lane Market
What: Probably the most famous of all London markets, this large Sunday market offering a variety of goods, with a bias towards clothing, particularly leather coats. There is a much smaller market in one street weekday lunchtimes.
Where: Middlesex Street and Wentworth Street, London E1
Nearest tube / rail: Liverpool Street, Aldgate East or Aldgate

Who: Portobello Road Market
What: the world's largest antiques market with lots of antiques, Victoriana and pseudo-antiques. If you're going to go, go early on a Saturday, because it gets very crowded.
Where: Portobello Road, W11
Nearest tube / rail: Ladbroke Grove, Westbourne Grove or Notting Hill Gate

Who: Smithfield Market
What: the Corporation of London's 800 year old meat market. Open Monday - Friday 04:00 - 10:00, come early in the morning and eat a fried breakfast at one of the local cafes with the market traders. A London institution.
Where: Charterhouse Street, London EC1
Nearest tube / rail: Farringdon or Barbican

Who: Spitalfields Market
What: once fruit and vegetable market, spitalfields is now a weekend attraction and organic food market. Go on a Friday or Sunday when the market is open from 10:00 - 17:00.
Where: Commercial Street, between Lamb and Brushfield Streets, London E1
Nearest tube / rail: Liverpool Street