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When Neil Armstrong and Buzz Aldrin descended to the moon in the Lunar Module, Michael Collins stayed up in lunar orbit in the Command Module.
Why did they need someone to stay behind?
At this point, rendezvous and docking with an unmanned spacecraft in orbit had already been done. Was there something else they needed someone up there for?
This is an outcome of several requirements and constraints:
[Space Task Group's] "Richard S. Johnston presented three demands: "shirt-sleeve" environment, three-man crew, and radiation protection. He discussed the need of the crews for a safe environment and for atmospheric control." (source)
The mission design switched to Lunar-Orbit Rendezvous (thanks to work by John Houbolt), and the Lunar Excursion Module could have the mass budget only for two astronauts.
"By the end of 1961, the newly named Manned Spacecraft Center had virtually swung over to the lunar-orbit rendezvous idea. Gilruth, Faget, and the other Apollo planners conceded that this approach had drawbacks: a successful rendezvous with the mother craft after the bug left the lunar surface was an absolute necessity, and only two of the three crew members would be able to land on the moon." (source)
The docking argument is not really valid since CSM's Apollo Guidance Computer (AGC) could receive commands directly from Mission Control and hence, could maintain fixed attitude required for docking without any intervention from the lunar module crew. (I read it somewhere in the handbooks - see http://www.ibiblio.org/apollo/links.html, see also letters from Grumman to ASPO cited here)
However, the overall doctrine during the Apollo program hinged on using crew members as backup for computers. If computers or communications failed, command module pilot would be instrumental in mission success. (A sextant, a slide rule, pencil and paper were provided for manual navigation).
"Shea and his staff reviewed these studies and presented the results to the rest of the manned space flight organization early in October [1962]. The contractors agreed that either two-man direct flight or earth-orbit rendezvous was feasible but both were less attractive than lunar rendezvous because the probability for mission success was lower, the first landing would be later, and the developmental complexity would be greater. The vote was still for three-man, lunar-orbit rendezvous." (source)
In addition to the above answers, the third crewman on the CSM performed visual checks of the LM after separation before descent, photography duties, regular purging of fuel cells, reconfiguring and relaying communications from the LM to mission control and platform alignments. Some of these may be possible from the ground.
I guess it was also a big comfort to have a guy on board just in case of some failure which would result in the loss of attitude or communication (and control) of the CSM. It's not just for computer backup. For example, a mechanical failure like some gas leak could push the CSM off attitude and through gimbal lock. That might be easily correctable by a guy on board, yet pose major problems to fix from the ground if communication was lost, and could prevent the LM from docking and kill everyone.
If the Lander had a problem, the CSM needed to be able to accommodate unusual docking circumstances. Sure, they had done unmanned docking before, but the computer on the CSM was not exactly great by today's standards.
