Amp-hour ratings are how manufacturers rate a battery's electrical capacity, in the same way you might rate a bucket's water capacity in gallons. Manufacturer amp-hour ratings are helpful when comparing different batteries and different brands, but be forewarned: all isn't necessarily what it appears to be on the label.
What is an Amp-Hour?
An amp-hour -- properly "ampere-hour" -- is a unit of measure describing electrical charge capacity. If a battery is said to have "20 amp-hours of capacity," it means that it can hypothetically discharge one amp of current for 20 hours, 20 amps for one hour, or any combination of amps and hours that equal 20 amp-hours. That might be five amps for four hours, 10 amps for two hours, or 6.66 amps for three hours.
Run-Time for Accessories
In theory, you can calculate the run-time for any of your battery's accessories by dividing the amp-hour rating of the battery by the amp draw of the accessory. You might have to convert from watts first. For instance, if you want to know how long you can bang a 1,000-watt stereo system on a heavy-duty 100-amp-hour battery, start by dividing the wattage by the voltage -- 12 volts, in this case -- to arrive at an 83.3-amp draw for the stereo system. Divide the battery's 100-amp-hour capacity by 83.3, and you get 1.2 hours, or about 1 hour and 12 minutes of tunes before the battery dies.
Manufacturers test and rate batteries according to a certain discharge time -- in the automotive industry, usually 20 hours. In theory, that doesn't affect the actual amp-hour rating, but it does make a difference on how long your battery will last if you consistently discharge it outside of that 20-hour parameter. If you're using a standard battery rated for a 20-hour discharge, and you routinely kill it in an hour with a banging stereo, or 150 hours with a tiny trunk-light bulb that won't shut off, then the battery won't be long for this world.