Though it's recently been rather overshadowed by the showier Tate Modern, Tate Britain (named after William Tate, the sugar magnate, who founded it) beside the Thames at Milbank, is still the biggest hitter in the London art world, with a fantastic permanent collection. It's particularly good for Turner and Blake, each of whom have a gallery to themselves, but there are also plenty more top international names, including modern British artists such as Tracey Emin, Francis Bacon and John Latham. Tate Britain is well-known for its temporary shows, the most notorious being the annual Turner Prize exhibition, which displays the work by four young artists and can be guaranteed to baffle and infuriate the public. Every three years a guest curator produces an overview on the current British art scene, which is always intriguing.
Tate Modern, just down the river from Tate Britain at Bankside, opened in 2000, and has been a great success, acclaimed both as a strikingly bold building - its origins as a power station are still obvious in the hulking brick structure - and for its imaginative exhibitions. The permanent display is divided thematically, and includes most of the big names of the twentieth century, including Monet, Rothko, Matisse and Picasso among many others. There are also temporary shows and there's always a specially commissioned piece in the Turbine Hall; it's fascinating to see how artists have responded to this gigantic space.
Note that both Tates, like all public galleries, have free entry (though temporary exhibitions are charged), and that there's a very handy boat ('Tate to Tate') that runs along the Thames between the two Tates, allowing you to visit both galleries in a day.
At the heart of London in Trafalgar Square, with its domed portico facing Nelson's Column, the National Gallery has to be the best located of Britain's art museums. Its collection, housed in a magnificent set of old-fashioned galleries and the bright new Sainsbury's Wing, ranges from the mid-13th century to around 1900, the Renaissance to Post-impressionism, with most visitor's highlights being Leonardo da Vinci's Virgin of the Rocks and Van Gogh's Sunflowers. There is also a noted collection of Impressionist masterpieces.
A number of smaller venues have excellent temporary exhibitions, generally of living artists. The Whitechapel gallery is in the east end, a long way from the standard tourist sites (though close to Aldgate East tube station). The Serpentine is small, but very well situated in Hyde Park. Both are free.
Larger temporary exhibitions can be seen at the The Royal Academy of Arts in Burlington House in Piccadilly. It was founded in 1768 with a remit to encourage British art, and it holds important temporary exhibitions - though note that it's not cheap at around ten pounds a ticket. Every summer, from May to August, it hosts an open exhibition - anyone can enter. The selected work is variable, and often rather conservative, but it's always worth checking out.
Finally, the Hayward Gallery, in the South Bank arts center, has three or four shows a year, generally surveys of trends in modern art, designed to complement the building's rather brutalist concrete architecture. Entry is around eight pounds.
For more artistic tourist attractions check out You2UK - your invaluable guide to getting the best out of a trip to Britain.