Greenwich is east of London proper and is also a very historic area. There used to be a Royal palace here and it was one of Henry VIII's favourites. The Royal Observatory is here now, high on a hill in the middle of a lovely park, and they also have a Planetarium, too. This is where the theory of longitude was hammered out over time and it is here where the Prime Meridian exists. You can stand with one foot on either side of the line and be in the eastern and western sides at the same time. The museum describes the journey to discovering and determining how longitude works and the difference it made for sailing the high seas.
At the base of the hill is the Queen's House and National Maritime museum in buildings dating from the 17th Century, designed by Inigo Jones. The Royal Naval College, designed by Christopher Wren is nearby as well, and is on the site where the Tudor Greenwich Palace stood.
In the town of Greenwich, you will find a market and lots of great little shops and pubs. There is also a small but unique museum, the Fan Museum. It tells you about the history, use and construction of fans and how they're made. There are some really nice examples on display as well as a very pretty cafe on site.
On the waterfront is the Cutty Sark, a merchant tea clipper ship. It was built in 1869. In 2007 a devastating fire broke out and the ship is undergoing restoration, due to reopen to the public in 2011.
Greenwich is an interesting part of London and isn't that difficult to get to. The DLR from the Tower station gets you there quickly and easily. The underground goes to North Greenwich on the Jubilee line. That is a bit further away from most of the attractions listed here but still in the general area. It will let you off near the big O2 venue (that used to be the Millenium Dome and is now a venue for concerts and other exhibits.) You can get a regular train from Victoria, London Bridge, Waterloo, Canon Street or Charing Cross. I think one of the best ways to get there is by the river. There is a river service from Westminster pier by Parliament which also stops by the Tower of London. It's a little over an hour from Westminster and about a half hour from Tower pier. There is a shuttle bus that will take you up to the Observatory from the DLR and pier station so you don't have to climb.