Okay, so, which is better? 800 mAh vs. 2000 mAh?

2000 mAh is better.

mAh is milliamp hours. From a 2000 mAh battery (when new) you should be able to (providing the manufacturer isn't lying) draw a sustained 1 milliamp current for 2000 hours.

mAh = 1/1000th of a Ah

ps3 60gb is rated at drawing 3.2 amps. If you had a battery that generated the required voltage and amperage for the ps3, and was rated at 2000mAh you would be able to run the ps3 for over 3/5 of an hour. *note that this is only theoretical as batterys are direct current while houshold power is alternating. The only reason i used the ps3 was that it was the only thing around me with an easy to read power consumption label.

Okay, so. When buying battery, shoot for the highest mAh...

A good rule of thumb is to shoot for the battery who's elements have the highest difference in electronegativity. Francium-Fluorine batteries ftw.

More colloquially, the further apart the two elements are on the periodic table, the better.