There are over 200 museums in London. Here's my pick. If you want to see a list of all look here.
I've written down each museum's website, the nearest Underground station and a reference on a Google map to make it easy to find. There is also a price for paid museums, but it is approximate so that you know how to consider it. The price may have gone up in the meantime, but discounts may also apply to children, families, students and seniors (most of the time, feel free to ask).

National free museums

National museums are free in London (donations may be collected and special exhibitions may be chargeable).

British museum

piltTottenham Court | Holborn  pilt

The museum is huge. If you watch every exp for 15 seconds, it would take you a few years without days off to watch everything. Special emphasis is placed on the historical findings of ancient Egypt, Greece and Rome. There is no hope of seeing everything, but the overall impression is powerful.

Natural History Museum

pilt South Kensington  pilt

Everything possible related to nature. From dinosaur skeletons to butterflies.

National Gallery

pilt Charing Cross  pilt

Over 2,000 paintings from the Middle Ages to the present day. Da Vinci, Botticelli, Caravaggio and Van Gogh.

Science Museum

pilt South Kensington  pilt

Everything related to science. Reconstruction of the first (mechanical) computer, Apollo lunar landing capsule, Watt's steam engine. Lots of interesting things for children too. My favourite.

Victoria and Albert Museum (V&A)

pilt South Kensington  pilt

Over 2.5 million artworks and design objects. One of the largest collections in the world. Fashion, furniture, sculptures, jewellery and more.

Tate Modern

pilt Southwark | St Paul's  pilt

Modern art. It is located in a converted building of an old power station. Suitable for a visit together with a visit to St Paul's Cathedral, as they are connected by a pedestrian bridge - the Millennium Bridge.

Museum of London

pilt St Paul's | Barbican  pilt

Everything about London from ancient times to the present day. Interesting stuff, but with an ugly messy structure. You want to go see something according to the sign, but it leads to a dead end corridor or back to where you came from.

Imperial War Museum

pilt Lambeth North | Elephant & Castle  pilt

Facts about wars from WWI to the present day.

National Maritime Museum

pilt Cutty Sark for Maritime Greenwich  pilt

Maritime history. Old maps and ship models. Admiral Nelson's uniform as he wore it at the Battle of Trafalgar.

V&A Museum of Childhood

pilt Bethnal Green  pilt

Toys, dolls, board games - everything related to childhood from 1600 to the present. There are also indoor attractions that children can participate in.

Sir John Soaneā€™s Museum

pilt Holborn  pilt

Home of a famous architect. Lots of paintings and antiques.

Guildhall Art Gallery and Roman Amphitheatre

pilt Bank | St Paul's  pilt

Contains an old amphitheatre found during excavations when the gallery was built.

Wallace Collection

pilt Bond Street  pilt

18th century French art. Titian, Canaletto and Rembrandt. The building itself is also worth seeing.

National Portrait Gallery

pilt Charing Cross | Leicester Square  pilt

The largest collection of portraits in the world.

Tate Britain

pilt Pimlico  pilt

British Art since 1500.

Geffrye Museum

pilt Old Street  pilt

Changing British homes through the ages. Each room is decorated in the style of a different era.

Wellcome Collection

pilt Warren street  pilt

Science, medicine, psychology. Darwin's walking stick can also be seen. :-)

Museum of London Docklands

pilt Canary Wharf  pilt

The history of London as a port city.

Other museums

Clink Museum

  >=£7.50pilt London Bridge  pilt

The old prison. Quite a terrifying place where you can get a picture of yourself behind bars as a souvenir.

London Transport Museum

 >=£17.50pilt  Covent Garden pilt

Vehicles, photos, posters, sound recordings, maps, uniforms, badges, films and videos. Everything related to transport in London.

Design Museum

  >=£23pilt  High Street Kensington pilt

Any design from cars to three-legged stools.

Royal Air Force Museum

  freepilt  Burnt Oak (far) pilt

All about the British Air Force.

Churchill War Rooms

  >=£19pilt  Westminster | St John's Park pilt

Underground spaces from where the country was run during the war.

Madame Tussauds

  >=£29pilt  Baker Street pilt

World famous wax museum. Several statues are made from "originals" because Madame got her hands on the severed heads of many celebrities.

HMS Belfast

  >=£16;pilt  London Bridge pilt

Battleship, anchored forever.

Charles Dickens Museum

  >=£9pilt  Russel Square | King's Cross St Pancras | Chancery Lane pilt

Over 10,000 items related to the famous writer.

The Queen's House

  >=£10 (with a guide)pilt   pilt

The infamous Henry VIII was born in this house.

Bank of England Museum

  freepilt  Bank pilt

History of banking and money.

Jewish Museum London

  >=£8.50pilt  Mornington Crescent pilt

History of the Jewish People. Judaism. The Holocaust. And many more...

Kensington Palace

  >=£15.50pilt  Queensway | Notting Hill Gate pilt

You cannot see the whole palace, only a part is open to visitors.

Sherlock Holmes Museum

  >=£15pilt  Baker Street pilt

"Museum" of a famous book character.

London Sewing Machine Museum

  Freepilt Tooting Bec  pilt

Unfortunately, it is only open on the first Sunday of every month. :-(

Royal Academy of Music Museum

  freepilt Baker Street  pilt

An incredible number of musical instruments. Also documents, pictures... Many famous Cremona string instruments and if you're lucky, you can even hear some of them.

The Postal Museum

  >=£10pilt Chancery Lane | King's Cross St Pancras   pilt

Postal matters. A lot is especially designed to make it interesting for children.

The Old Operating Theatre and Herb Garret

  >=£6.50pilt London Bridge  pilt

A 19th century operating theatre located above the church. Various surgical techniques are demonstrated.

Museum of Brands, Packaging and Advertising

  >=£9pilt Latimer Road  pilt

Over 12,000 parcels, boxes, cups... from the era of Queen Victoria to the sixties of the 20th century.

Magic Circle Museum at the Centre for Magic Arts

 only by special agreementpilt Euston Square  pilt

Harry Houdini and others...

London Museum of Water and Steam

  >=£12.50pilt South Ealing (far enough)  pilt

Steam engines and locomotives. If you want to see these in action, plan your visit for a day off or a Bank Holiday.

Anaesthesia Heritage Centre

  freepilt  Regents Park pilt

More for medical enthusiasts. All about pain and painkillers.

Brunel Museum

  >=£3pilt  Canada Water pilt

Marc Isambard Brunel and his son Isambard Kingdom Brunel were the engineers who first began to build large tunnels under the river.

London Canal Museum

  >=£5pilt  King's Cross St Pancras pilt

London's canals, shipping and freight on these.

The Cartoon Museum

  >=£7pilt  Tottenham Court Road pilt

Animated films and comics.

The Cinema Museum

  >=£10pilt  Elephant & Castle pilt

Everything from the cinema. Just from the cinema, not from the films. Posters, decor, seats and even bits of carpets.

Dennis Severs' House

  >=£10pilt  Liverpool Street pilt

18th century house. Its ten rooms have an air of the past, as if the residents had just left here.

Fan Museum

  >=£4pilt  Cutty Sark for Maritime Greenwich pilt

Fans from the 11th century. Located in two nice 18th century houses.

Freud Museum

  >=£7pilt  Finchley Road pilt

Sigmund Freud settled in this house in 1938, having fled from the Nazis in Vienna. The entire interior has been preserved, including the famous sofa.

Kirkaldy Testing Museum

  >=£5pilt  Southwark pilt

Museum of Engineers and Technology Enthusiasts. Among other things, a 116 ton fully operational 'test machine'.