This is a great guide that will teach you all you need to know about Lipo's:
http://www.rchelicopterfun.com/rc-lipo-batteries.html
The "C" rating has nothing to do with temperature, it's an indication of how much current the battery can deliever (or receive, when you're talking about charging). The battery never "pushes" the electricity to the system, the system draws it out of the battery.
The "20C" number you see on the stock battery (for example) refers to how fast it can supply electricity to the system. Increase in that current draw occurs when the motors or other items are revved up or increase their demand. Batts need to be able to supply the current at or above the rate the system draws it. Batts labeled "30C-60C" mean they can deliver 30C under normal conditions, but up to 60C in short bursts.
1"C" = the battery's mAh divided by 1000. Therefore, 1C for a stock Phantom battery is 2.2Amps. 2C = 4.4Amps, and so on. Batteries can be charged at 1C or multiples of that, but only those that are rated for 2C or higher charging should be done at those values.
I charge my Phantom batteries at 1C (2.2A), but I think they can handle 2C. The MadDog 2700mAh batts are rated (I believe) at up to 5C, which would be 13.5A, but I still charge them at close to 2C usually (5Amps).