The gross calc were done on earth, with the actual firing - the button pushing - beiong done locally. (For example, the retro rockets were fired by the astronauts returning to earth from earth orbit, the lunar orbit firing was done behind the moon so it could not be controlled from earth.)
A little bit dramatic, but there were a few astronaut controlled firings: look at the Apollo 13 hand controled burn to orient it back towards earth. There, they aimed by controlling the attitude of the capsule by aiming the window of the LEM at earth. The timing of the burn (how long the rockets were fired) was predicted on earth (fire for 33 seconds, for example, at 1432 hours).
The Apollo onboard computers were 15bit plus parity machines running at about 40,000 ops/Sec. They had about 38KB of core memory, about 2K of that writable, the rest was stored data and programs.
About the same as an HP-41CX. You could fit it into a watch, but I don’t know of any generally available watches with this sort of general purpose computing ability (if I did, I’d probably own one. ;) )
http://history.nasa.gov/afj/compessay.htm
http://www.ibiblio.org/apollo/