Good question, moved up to Selecting a Handgun.
The things that break depend on the model of firearm. What one can fix also depends on the gun.
Small springs, extractors, firing pins, magazine release assemblies are what I have seen break the most. Usually those things are not to hard to learn to fix.
I'm sure there are others on par but one of the most reliable and also easy to fix guns is a Glock. Here is a good video on how to detail strip one.