Acceleration is the rate at which an object changes its velocity -- for example, the increase in a skydiver's speed after he jumps out of an airplane. Acceleration is a vector quantity, which is a quantity that takes into account magnitude and direction.
Acceleration does not necessarily have to do with fast movement, as an object moving quickly at a consistent rate is not accelerating. Rather, it has to do with increasing the speed of movement. An example is a car stopped at a red light. When the car begins to move again, it is accelerating. When the car eventually slows down, it also undergoes acceleration, but this time, the acceleration is in a direction opposite to its movement.