A batteries Discharge rating (C) is a measure of how fast a battery can be drained safely without damaging it.
(C) is equal to the number of times a battery could be discharge in an hour safely. 1C means you can discharge the battery in 60 minutes, while 60C means you can discharge it 1 minute.
By itself (C) doesn't mean much unless you also know the mAh(milliamp hours). A battery with 1000mAh and 10C can put out 10 amps in 6 minutes. A battery with 2000mAh and 10C can put out 20 amps in 6 minutes! A battery with 1000mAh and 20C puts out 10 amps in 3 minutes while a battery with 2000mAh and 10C can also put out 10 amps in 3 minutes while still having 10 amps left to go.
A battery's A(Amps) can be determined by converting mAh to Ah(amp hours) by diving by 1000
(2000mAh/1000) = 2Ah
Then multiply the Ah by the C rating
(2Ah x 10C) = 20A
So a 2000mAh 10C battery can be out 20A in 6 mins!
Constant discharge is for how fast it can be discharged while being constantly drained like a flashlight or someone shooting in full auto for very long bursts. Burst discharge is for how well it can keep its voltage up while being started and stopped similar to how airsoft aegs are used in semi auto and short bursts.