London's newest gallery, the Tate Modern (so called to distinguish it from the Tate Britain across and west at Millbank -- along the river Thames towards Pimlico and Chelsea, which houses traditional and classical art) opened on Bankside in May 2000. It's in a majestic position, virtually right opposite to St Paul's Cathedral and to the Millennium ("bouncy") Footbridge. Even if modern art is not your thing, the museum itself is still worth dropping into for its extraordinary architecture and because there is normally an interesting exhibition in the main foyer, formerly the turbine room.
With that introduction, it won't surprise you to hear that the museum is housed in what was previously called the Bankside Power Station, originally designed by Sir Giles Gilbert Scott (also the architect of the Battersea Power Station - you will have seen its four chimneys from the London Eye), the Liverpool Anglican cathedral, University libraries in Oxford and Cambridge, Waterloo Bridge, and the famous British red telephone box (sadly fewer of those to spot these days)).
The power station itself was a brick-clad steel structure of c4.2m bricks and its central chimney stands at 99m (325ft), apparently to ensure that it didn't top the dome of St Paul's; it replaced a coal-fired power station in 1952 and expanded in 1963 but, by 1981, oil prices had risen such that the power station was uneconomic compared to other forms of electricity generation. It stood unwanted until 1994, when the Tate Gallery realised its collection had outgrown its Millbank home and took an option on the site. It took until 2000 for the Thameside megalith to be converted, forming part of the Southbank regeneration which included the Globe Theatre, Vinopolis, the Clink ghoul-show and restaurants near the NT.
Most striking is the turbine room, which operates as a "covered street" (3,300 sq m - 35,520 sq ft) to show works of art -- often one enormous piece, which could not be accommodated elsewhere, fills the entire expanse. The whole museum (on 6 floors) has a total internal floor area of 34,500 sq m (371,350 sq ft) and numerous modern art exhibitions (some permanent and some travelling) which span paintings, cinema, interactive art, and sculpture to satisfy the most inventive and contemporary of tastes in art. (If your taste runs to the more traditional, also try the National Gallery on Trafalgar Square or the Tate Britain (as above -- both free).
Open Sun-Thurs: 10am-6pm; Fri-Sat: 10am-10pm (save 24-26 December). Admission free - donations encouraged. Obviously a nice route is along the South Bank on foot, but on a fine day, also try a boat trip -- the Tate to Tate runs every 40 minutes along the Thames between Tate Britain (Millbank -- see above), London Eye, and Tate Modern.