It's not an exaggeration to say that the British Museum is one of the World's greatest museums. Containing exhibits from everywhere on the planet and delving into the history of mankind from 3 million years ago up until yesterday (and sometimes even tomorrow), the BM is one of the absolutely essential sights to see in London.
Here we have everything. The British galleries will give you a fantastic grasp of British history that is unrivalled by any other on the planet, while the world galleries will show you everywhere on earth, all within one very easily accessible (and also extremely free!) place. Go during the week if you can - it will be much quieter.
Try the back entrance (Montague Place) to get in quickly - a cloakroom there will take your bags. Take a packed lunch and plan for all day!