ahh..reminds me of circuits from 3 years back.
depending on your battery, you can test it a few different ways. if it is a NIMH or NICAD battery, its easier to test it with a digital camera. the most laymen way to test it would be to see how many pictures you can take with flash, and compare it with another set of batteries that have a known rating (1600mAh) and a known number of pictures you can take before camera doesnt work. now this method isnt highly scientific but it is the easiest way to gauge approximate battery capacity.
for an alkaline battery, you have to sacrifice the battery to run tests on. best is to read up on manufacturers datasheets.
comprehensive data on energizer
the more scientific way to do it...would be something like the following.
1) place ammeter (multimeter) in series with battery and electrical device like a bright flashlight or anything that can deplete the battery to low levels in 8 hours or less.
2) measure current draw (in mA) every half hour till device ceases to function acceptably.
3) graph current draw versus time. area under graph is approximate mAh rating.
now..if you chose to 'apply' a current and watch the voltage of the battery, you must do some other things. you can not be sure of what the current is at any given moment in time. current is not constant with a fixed resistance and variable voltage as you know from V = IR. you can however, watch the voltage drop and measure the resistance of the device to calculate the current. you can chose to measure voltage across battery under load or current draw under load depending on your setup.
if you want to know in detail what the graphs may look like..check out the datasheets i linked in this post. they have graphs for all batteries under constant resistance, constant current, or different temperatures.