The second derivative is the derivative of the derivative.

notations include:

Physical interpretation

If f represents position, and f' velocity then f'' corresponds to acceleration(rate of change of velocity). 

Ball example

Intuitive interpretation

f''<0 means the graph of f curves down. (think of -x².)

f''>0 means the graph curves up.(think of x².)

The third derivative is the derivative of the second derivative.

notice these facts