The Tate Britain is the home of the UK's painting collection from 1500 - present. In 2000 they moved their (huge) collection of modern art (Impressionism and later) to the newly established Tate Modern, so that the Tate Britain - "this" Tate - now holds all the Old Master and British Master paintings. Which makes it *my* kind of museum!
It's got an unbelievable collection of JMW Turner paintings... it must be the largest collection of his work anywhere. Gallery after gallery, dozens of canvases representing every phase of his (long) career. His late work just oozes atmosphere and emotion. This image (below) of a horse and skeletal rider was my favorite... I've always equated Turner with landscapes and found this evocatively terrifying work a very pleasant surprise.
There are also quite a few Francis Bacon paintings, and good ones at that; as well as a surprising number of John Singer Sargent pictures, considering we Americans like to think of him as American. There are the requisite British artists you expect to see... the Royal Academy favorites like Gainsborough, Hogarth, Reynolds, Raeburn, Romney, etc. Equally impressive sculptures, architecture (the museum itself) and works on canvas here... along with the fact that (like all the other major museums in London) it's free, make this a 'must see' attraction in my book.
Honestly this museum is worth a trip for the Turners alone. They've got some unbelievably beautiful and experimental canvases that'll just knock your socks off.