Many London museums do not require a ticket and admission is free. But that's not all. Many museums also offer free tours. Some museums offer a tour for a few pounds. Most museums also have audio tours - a great way to explore London museums on your own, but get maximum information.
Some of the places to visit are:
The British Museum – free admission and free tours.
The British Library – free admission, and free events.
The National Gallery – free admission and free tours.
National Portrait Gallery, with audio tours available for £3.
Museum of London – free admission, museum tours require a ticket.
Imperial War Museum – no charge for admission to the museum, some tours and talks are free, others are at extra cost.
Royal Air Force Museum – also a museum with free admission, and a great number of events and exhibitions (some might require an admission fee).
Saatchi Gallery – admission is free of charge, check details for talks and workshops.
Science Museum – no entrance fee to the museum, some exhibitions might be extra. Check Science Museum in London calendar of events.
Tate Britain and Tate Modern both have free admission. Check their websites for events, tours and exhibitions – there are many.
Victoria & Albert Museum is one of the most interesting museums not only in London, but in the world. No tickets are required to enter, the museum is free, except for some exhibitions. Check their What's On section to learn about ongoing and future events and exhibitions.