30A ESC stands for 30 Amp Electronic Speed Controller. It's an electronic circuit that controls the speed and direction of a brushless motor. It can also act as a dynamic brake. 
30A ESCs are designed for quadcopters and multi-rotors. They can drive motors that consume up to 30A current. They work on 2-3S LiPo batteries. 
30A ESCs have the following features: 
They can vary the speed and direction of a brushless motor.
They can act as a dynamic brake.
They have an onboard BEC that provides regulated 5V (2A max draw) to power the flight controller and other onboard modules.
They have a constant current of 30A and a max current of 40A (>10s).
They have an input of 5.6V - 16.8V.
They have a super high refresh rate, no buffering of the input signal, and a response rate of more than 490Hz.
They have a motor frequency of 16KHz, which gives the fastest response of the motor and quietest operation.
ESCs are normally rated according to maximum current. The higher the rating, the larger and heavier the ESC tends to be.