An Amp-Hour (ampere-hour, Ah) is a way of describing a battery's capacity - how long it will run before it is drained down. But I can be be more specific: the amp-hour rating for a given battery is the maximum amperage that can be drawn continuously until the battery is completely discharged, flat, empty, drained, dead, over a specific time period.
How Are Amp Hours Determined?
Battery manufacturers complete tests on their batteries to give them an Amp-Hour rating. A typical time period for a test is 20 hours, but it varies - batteries are tested over different periods, such as 24 hours, 75 hours, even 100 hours. But as you'll see below, a 100 Amp-hour battery tested over a 100 hour period (i.e. 1 amp drawn for 100 hours) will not have the same capacity as a 100 Amp-hour battery tested over a 20 hour period (i.e. 5 amps drawn for 20 hours). It turns out that if you drew 1 amp continuously from the 20 hour test battery, it would last for 110-120% longer than on the 100 hour test battery. I'll (try to) explain that later.
Let's Do an Example Calculation
Suppose you have a 100 Amp-hour battery, tested over a 20 hour period. 100 Amp-hours divided by 20 hours = 5 amps. That means that the battery manufacturer claims the battery can sustain a 5 amp load for 20 hours until the battery is completely dead.
That's great, but now put it into realistic terms. For starters you don't want to drain a battery to it's completely flat, empty, dead state as it will significantly reduce the life of the battery. A good rule of thumb (battery type depending) is that you should only drain a battery down to 40% of it's original capacity, or in other words you've only got 60% of the battery to use. This number should be stated by the manufacturer in the documentation.
Back to the 100 Ah, 5 Amp load example: We said you had 20 hours of time when drawing 5 amps, but in fact you only have 60% of that time, so 20 hours x 60% = 12 hours.
Up until now, what we have said is true because we have used the same amperage over the same time period at which the battery was originally tested (20 hour rate). So what if you took your 100 amp-hour battery, but wanted to draw 10 amps from it until it was at the safe discharge level (60%)?
Do the math: 100 Ah / 10 Amps x 60% = 6 hours, right?
The More Amperage You Draw, the Lower the Battery Capacity
That's right, the effective amp hours available drops with the more amps you draw. All of a sudden that 100 Amp-hour battery is not what it says it is. Ok, back to the numbers, lets try to keep this clear:
Using the 100 Ah, 20 hour test battery, at 5 amps draw, you get the full 100% of the battery rating (that's how the battery was tested in the first place, so you should get all 100%!). But step up the amperage to 10 Amps, and you will lower the capacity of the battery by about 10% (or 90% remaining).
That makes 100 Ah x 90% = 90 Ah / 10 Amps draw = 9 hours x 60% (max safe discharge number) = 5.4 hours. And that's quite a difference from the 6 hours you thought you had.
Where Did I get the 90%?
I'm going to stop here and leave you hanging - except to say that there's a relationship defined called Peukert's Equation, and that's where the approximate 90% comes from. The Peukert Equation quantifies the above phenomenon so that you can predict how much time you'll actually have on a battery given a specific discharge rate.
What's Next - How About A Helpful Tool?
My guess is that if you're still reading this your about ready to start figuring out how many batteries you need, and you could really use a little help with the calculation. You might have gone to the Wikipedia site above to investigate what Peukert had to say, but exponents aren't really your thing, and you can't find your calculator anyway.