Why voltage drops when I short a battery?
When I short an AA battery, I have observed that it's voltage drops a
  couple hundred millivolts


If you short a battery it produces zero volts at the terminals. If instead you place a heavy load on the battery, it will produce a significantly lower terminal voltage because, you cannot expect a battery to be able to supply infinite current.

Every real and practical voltage source has internal series resistance (also a bit of inductance too). It is the internal resistance that limits the short circuit supply current. On any load this internal resistance forms a potential divider with the load resistance and the terminal voltage will drop.

Regarding the voltage "slowly" returning to its full value following a short circuit, it's a chemical process in the battery and shorting it out, causes an overload to that chemical process and it takes time to recover.

Wiki article - recovery effect