Ah means Ampere Hours. All batteries are rated according to their power output. Virtually all car batteries run at 12 volts for but the car battery Ah and CCA ratings vary.
The average car battery will have an Ah rating somewhere around 50Ah. The 50Ah means that it can push 12 volts out at a rate of 50 amperes in an hour.
For example : If a light bulb used only 1 ampere in an hour then a 50Ah battery could light up 50 light bulbs for an hour. Alternatively it could power 1 light bulb for 50 hours*
Your car uses the battery for 2 main purposes (1) the Engine (2) the Electrics such as lights, wipers, satellite navigation, stereo and so on.
The engine size is primary. A larger engine uses a larger starter motor and needs more battery power to turn it over first thing of a cold morning. Other factors include the number of cylinders the engine has, each of which needs converted energy to create a spark for combustion.
With the car electrics - most modern cars have sat-nav and a fairly good stereo. 20 years ago this would not have been the case. Additions such as a powerful stereo were usually added later and sat nav's didn't even exist. In doing so the driver 20 years ago would probably not have found his or her battery going flat prematurely because the rest of the vehicle had less electronics on-board.
These days the story is quite different and modern cars are quite power-hungry. The Ah number matters more because cars are becoming more like computers in their complexity, they do a lot more thinking. Clear examples of this are 'engine management systems' , 'anti lock braking computers' , 'traction control' , cars that have buttons such as 'sport mode' and so on - each of which has its own micro-processor that measures the car's environment.
Can I use a battery that has a smaller Ah rating than my previous battery ?
In the vast majority of real world situations the answer would be NO.
Can I use a battery that has a larger Ah rating than my previous battery ?
Yes provided the figure doesn't stray much more than 10% over.
What if I cannot find the Ah number on my battery ?
See our article called 'Which car battery do I need - Part 1'
*(!) PLEASE NOTE : Car batteries are designed to be constantly re-charged by the car alternator. If you exhaust a car battery and do not recharge it, the innards will degrade quickly ( see HERE for a fuller explanation )