A-H
Brian, in general, amp-hours is a broad specification to tell you how much power a battery can hold before going dead ("flat" if you're on the other side of the pond).
If you divide the amp-hour (A-H) rating by the number of amps you intend to draw, the result will be how much time the battery will last. If you divide the A-H rating by the time you want the battery to last, the result will be the maximum current that you can draw.
Reversing all that, if you need to run something at 3 amps for a period of 20 hours before recharging, you'll need 3 x 20 or 60 A-H of battery capacity.
Now, this is not a linear equation, but general only. In other words, don't expect a 60 A-H battery to put out a maximum of 3000 amps for 20 milliseconds (its internal resistance will limit maximum current) or for one to last for 6900 years at a draw of 1µa (self-discharge will insure that it won't even hold a no-load charge for that long). And that curve varies for different battery chemistries.
Milliamp-hours is a similar unit of measure. A battery rated for 650mAH is the same as one rated for 0.65 amp-hours.
Dean