Since this is a particularly confusing part of measuring batteries, I'm going to discuss it more in detail.
Power capacity is how much energy is stored in the battery. This power is often expressed in Watt-hours (the symbol Wh). A Watt-hour is the voltage (V) that the battery provides multiplied by how much current (Amps) the battery can provide for some amount of time (generally in hours). Voltage * Amps * hours = Wh. Since voltage is pretty much fixed for a battery type due to its internal chemistry (alkaline, lithium, lead acid, etc), often only the Amps*hour measurement is printed on the side, expressed in Ah or mAh (1000mAh = 1Ah). To get Wh, multiply the Ah by the nominal voltage. For example, lets say we have a 3V nominal battery with 1Amp-hour capacity, therefore it has 3 Wh of capacity. 1 Ah means that in theory we can draw 1 Amp of current for one hour, or 0.1A for 10 hours, or 0.01A (also known as 10 mA) for 100 hours.
For example, in this image, a coin cell can drive a 3.9Kohm resistor and provide 230mAh (which is what its rated for) before dropping to 2V, but if its a 1Kohm resistor, it will only provide 125mAh (image from http://biz.maxell.com/en/product_primary/?pci=9&pn=pb0002)
The way the power capability is measured is in C's. A C is the Amp-hour capacity divided by 1 hour. So the C of a 2Ah battery is 2A. The amount of current a battery 'likes' to have drawn from it is measured in C. The higher the C the more current you can draw from the battery without exhausting it prematurely. Lead acid batteries can have very high C values (10C or higher) , and lithium coin cells have very low ones (0.01C)