The Royal Observatory was established by Charles II in 1675 for the purpose of observing the motion of the heavens and to help perfect navigation. It is the location of the Prime Meridian (dividing the world into east and west) and the source of Greenwich Mean Time (GMT) - a time ball still drops at precisely 1pm each day to help sailors and others set their time pieces correctly. The Royal Observatory moved to Herstmonceaux Castle in Sussex in 1957 - what remains is essentially a fascinating museum. Also on site is the Peter Harrison Planetarium.
Part of Royal Museums Greenwich, which includes the National Maritime Museum, Cutty Sark and Queen's House. It's about a 20 minute walk from the nearest railway station (Cutty Sark DLR or Blackheath main line) and the museum is located in the middle of Greenwich Park on a steep hill with no parking; check the website for more details.