Amphour 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 amphour 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 AmpHour?

An amphour  properly "amperehour"  is a unit of measure describing electrical charge capacity. If a battery is said to have "20 amphours 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 amphours. That might be five amps for four hours, 10 amps for two hours, or 6.66 amps for three hours.
RunTime for Accessories

In theory, you can calculate the runtime for any of your battery's accessories by dividing the amphour 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 1,000watt stereo system on a heavyduty 100amphour battery, start by dividing the wattage by the voltage  12 volts, in this case  to arrive at an 83.3amp draw for the stereo system. Divide the battery's 100amphour capacity by 83.3, and you get 1.2 hours, or about 1 hour and 12 minutes of tunes before the battery dies.
Caveat

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 amphour rating, but it does make a difference on how long your battery will last if you consistently discharge it outside of that 20hour parameter. If you're using a standard battery rated for a 20hour discharge, and you routinely kill it in an hour with a banging stereo, or 150 hours with a tiny trunklight bulb that won't shut off, then the battery won't be long for this world.