Getting to grips with battery C ratings
Batteries that are designed for cyclic use (ie discharge and then recharge) generally have a stated capacity based on normal expected use. A battery for off-grid power, for example, is normally rated at C100, which indicates how much capacity (ah or amphours) the battery could supply if it was being drained at 1/100th of the capacity per hour. For applications where a battery is likely to be drained more quickly (the battery running your camping fridge, for example) it is usual to quote a C20 figure, where the battery is expected to be drained at 1/20th of its capacity per hour. You'll also see figures of C10 (1/10th per hour), C5 (1/5th or 20% per hour), 1C (100% in an hour), or a multiple of that. If a 2v battery for off-grid use has a C100 rating of 1000ah, it might have a C5 rating of 700ah, indicating that if it is drained more quickly than normal, the voltage will drop more quickly and, under load, the battery will hit rock-bottom more quickly too. So the quoted amps for a given C rating for a battery allows you to make a direct comparison with other batteries of a similar type for a similar purpose, assuming that the battery manufacturer has been honest with the assessment and testing, and assuming the batteries are of similar standard. The quoted ah capacity and C rating doesn't tell you anything about quality or longevity!
Importantly, the C rating doesn't tell how fast a battery could deliver power if needed and if pushed, without damage, and some batteries (eg car starting batteries) don't, generally have an ah and C rating at all. By way of example, a 50ah gel battery might be used for an electric outboard (trolling) motor, and the motor will, likely, pull around 50amps at top speed, so the battery is going to be running at 1C - discharge within an hour. The 50ah rating is at C20, a much more gentle pace, so at full speed the battery won't last an hour, and might only last 30 or 40 minutes, but at mixed throttle settings it might last 3 or 4 hours - we suggest taking oars along if you are going to test this to the limit! Also, 1C is ok for a good gel battery, but 3C is pushing it, so there's not much point in buying a battery that is too small. Aim for at least the same capacity as the quoted thrust, in pounds, of the outboard.
Car batteries - well a typical starting battery of about 50ah C20 capacity will be able to deliver about 500 cold cranking amps for a few seconds, so it is working at about 10C (ie it'd last 1/10th of an hour but, because it is being pushed at around 200 times what it would be rated at if it was a deep cycle battery, there's no chance of it cranking the car for 6 minutes and, in reality, anything over a minute is probably pretty good going). The plates in the battery are designed to handle this. If you use a small deep cycle battery for the same job, it might start bending itself, inside, as it tries to cope.
The lithium polymer (lipo) batteries used to power the motor in electric radio control planes, by comparison, are very highly engineered and typically around 30C (ie designed to exhaust themselves in 2 minutes if really pushed), with some offering capability up to 80C and beyond - from full to empty in less than a minute - if you are prepared to pay for it. The bi-product of all that energy is heat, and a certain instability and risk of failure (with dramatic results), but it shows how different designs and materials create batteries suited to different applications.
Finally, a firework or a stick of dynamite is not quite a battery - a power storage device - but it is a fuel source designed to release power, just once, but very quickly indeed (with the help of a bit of oxygen etc). The C rating for them might be more like 36000C (please don't write in with alternative calculations though!).