Not much, by today’s standards. There’s an amusing write-up here on the SparkFun site about the Apollo Guidance Computer, comparing it to a contemporary microcontroller chip. I agree with the author that possibly the most interesting comparison is that at the time of its construction, there were probably only a few dozen people in the world who understood and had access to this sort of technology. Today, thanks in part to projects like the Arduino, anybody with a personal computer and a teeny budget (e.g., under $50 for the Arduino and some LEDs and stuff) can learn how to program and use a microcontroller.
The SparkFun entry draws upon material from a book titled Digital Apollo which looks like an interesting read.
Cool article. I remember when people marveled that there was more computing power in a Furby than the Apollo luner module.