I apologize in advance for the length. I kind of got on a roll :J
"C" refers to the battery's one hour capacity, and the C rating is C divided by the discharge rate in hours. The 'C rating' is used as a measure to inform the user how the battery will perform at multiples of the one hour rate. Specifically for lithium based batteries, it allows you to determine the maximum safe discharge current that won't cause damage to the battery or fire.
C is specified in amp-hours (AH) or milliamp-hours (mAH) depending on battery size and the one hour rate is referred to as 1C. So a 3.0 AH battery is rated for a 1C capacity of 3.0 amps of load for one hour at which time you would expect the battery to be completely discharged. That same battery should be able to supply 0.03A (1/100th C) for 100 hours, or 300 amps (100C) for 1/100th of a hour. That is true theoretically but not practically. In fact, the battery will run longer than 100 hours at 0.03A and less than 1/100th hour at 300 A. Here's why.
Real batteries have internal impedance which can be looked at like a resistor in series with the battery. That impedance varies with current draw, state of charge, cycle and calendar age, and battery chemistry. The result is, the more current you draw, the higher the voltage drop in the battery. Some of that impedance is fixed, for example the resistance of the internal battery straps, but part of it is a result of the chemical processes in the battery and so it rises with current flow. The voltage drop gets turned into heat which can cause a catastrophic rise in battery temperature, not to mention very poor motor performance due to reduced terminal (output) voltage. As if that isn't trouble enough, the chemical process of discharging a li-poly battery is exothermic. This means the battery produces heat as a by-product of discharge, The faster you discharge it, the more heat you get. Finally, high current flow reduces the overall chemical efficiency of the battery and so all batteries deliver reduced current capacity with increasing load.
This is where we get the notion of the 'C' rating of lithium batteries. It is a means to inform the user of the maximum safe rate at which the battery can be discharged. That is only safe, not recommended, or high performing, or long battery cycle life, or anything else. A 100C rated battery should be able to provide 100 times it's one hour rate continuously without bursting into flames. But there is no standard for the percentage of current capacity you will recover at that rate, how much voltage drop you will suffer, what the cycle life will be, or how hot it will get. As if that wasn't confusing enough, some manufacturers grossly overstate the performance of their products making the ratings completely meaningless. I have seen some cheap '45C' batteries that struggled at 15C when brand new. Don't ever take a high C rating literally until it has been tested in your application.
Having said that, you really have to take this whole topic with a grain of salt. A 20C battery will discharge in, at best, 3 minutes, while a 60C battery will do so in a minute. I don't know about you, but I don't fly any drone with a runtime much under 15 minutes (4C), nor do I fly anything with a runtime under 3 minutes (20C). What is more important is the ability of a battery to provide short pulses of high current, which is necessary, for example, to keep a drone stable in hover. In those cases the high C rate voltage drop limits the amount of power and thus torque that can be delivered by the motor, even for a short time. Temperature rise is less of a problem as the battery temperature rise is a result of heat loss integrated over the entire discharge time; the short bursts don't have a large effect. A low or high impedance results in a punchy battery or a soft one. At the very least, a soft battery would require different tuning parameters in the flight controller than would be required for a punchy battery due to the limited motor torque and acceleration. At worst, it will limit the flight envelope in which the drone can operate.
As with anything, more $$ generally gets you better quality. Buy a battery from a known US brand and you will likely not have an issue. Get yourself an eB*y or A**zon special from an unknown manufacturer or seller, and you don't know what you'll get. Some are fine, some are terrible, and it's impossible to tell ahead of time which is which. The 'same' battery can even from batch to batch as some of these guys resell the cheapest cells they can lay their hands on, and the real manufacturer varies over time.
Happy flying :J
Electrical & Software Engineer (retired)