"C" is a rating that specifies how much current the pack can support in a discharge. The "C" rating multiplied by the pack capacity will give you a number in Amps.
A specific example is a "20C" 4000mAHr pack. So the max discharge should be 20*4000 mA (ignore the "H") =80,000mA or 80 Amps.
I note that an actual 10C discharge would completely discharge the battery in 1/10 of an hour, or 6 minutes. In the previous example, that would be a 40 Amp continuous discharge.
"C" isn't universally defined by the Lipo vendors--sometimes it seems to be a marketing gimmick.
You basically should know that the main limit on a Lipo is how hot it gets during a discharge, and that is what sets the "C" limit. Recall that heating goes as the square of the current (in Amps) * the internal pack resistance. A lower internal resistance pack will have a higher C rating, so that is the physical meaning of "C".
You can stop here and not read the rest!
Another detail is that you can't calculate the heating with "average" current--the current we typically measure with our watt-meters and data recorders. So for an example here, if you are flying at 75% throttle on the ESC and measuring an average current of 40A, the actual current is 40A/0.75 or ~54 Amps for 75% of the time, and 0 Amps for 25% of the time. The heat produced during the off period is just 0 watts, but the heat during the on time is (40/0.75)2 * Rint the internal battery resistance. However since this is only happening 75% of the time, the actual heating is just 402/0.75 * Rint. Notice this is 4/3 (just 1/0.75) times the heating you would have calculated using the average 40amps.
So looking at the 20C 4000mAHr pack, a 40A discharge would be a 10C discharge. However if you are pulling this at 75% throttle, you are actually heating up the battery pack as if you were making a 13.3C discharge. So partial throttle does exact a higher price on a pack than full throttle (where average amps=instantaneous amps).