Unless noted, all photos are from November 2010 (some are from July 1977).
St. Paul's Cathedral
St. Paul's Cathedral is the national church of England. It was built after London's Great Fire of 1666 and survived the blitz of World War II. No photography is allowed inside.
Westminster Abbey is where Kings and Queens of England are crowned and buried. Tombs of other great people are also here, including Geoffrey Chaucer, Charles Dickens, and Laurence Olivier. No photography is allowed inside.
Tower of London
The Tower of London started out with the White Tower (still the centerpiece), which was built in the late 11th century. It was enlarged over the centuries and used to be both a royal residence and a prison. The Crown Jewels are now housed there. Photogoraphy was not allowed in the Crown Jewels exhibit.
Churchill War Rooms
The Churchill War Rooms are the underground location where Churchill and his cabinet conducted business during World War II. Originally the rooms were not capable of withstanding a direct attack, so they were reinforced. In addition to seeing the war rooms, there's also a Churchill Museum which has a large collection of Churchill memorabilia.
The current Globe Theatreisn't the original one of Shakespeare's time, but is one which was built thanks to the efforts of actor Sam Wanamaker. The site of the original Globe is now an approach to a bridge crossing the Thames, so the current one is a bit to the northwest.
The theatre seats about 800, and an additional 700 can stand on the ground in front of the stage. The original Globe had the same dimensions but they could pack twice as many people in for performances.
The British Museum has a very comprehensive collection of many artifacts, including things which show the rise and fall of ancient Egypt, Greece, and Assyria. However, that just touches the surface; it would take several days to do justice to what the museum has.
Victoria and Albert Museum
The Victoria and Albert Museum displays a wide array of items from the ancient to the modern. It was named after Prince Albert and Queen Victoria, since they gave much support to the museum (which was named the South Kensington Museum at the time).
Along the Thames
University of Cambridge is easily reachable from London by train. Undergraduates are actually not admitted to the University, but to one of the colleges. In fact, the colleges are separate entities from the University as a whole, unlike most other universities (Oxford is also structured like Cambridge). While many of the lectures can have lots of students, each student has at least one hour a ewek dedicated (with one other student) with what could be thought of as an advisor.