In simple terms, the rate of change of velocity.
Just as velocity is the rate of change of position with time, acceleration is the rate of change of velocity with time, and like velocity, is also a vector quantity. It can also be thought of as the rate of change of the rate of change of position; how the change of position changes with time.
In everyday usage, the term "acceleration" probably means to "speed up" while driving, or increase something's speed. In the language of physics, the term is slightly more broad. Since acceleration is the rate of change of velocity, whether that be a negative or positive change, or a change in the direction of the particle, the term "acceleration" means either speeding up, slowing down, or changing direction. A particle can have a constant speed, but be changing direction, and still have an acceleration.