To accelerate an object is to change its velocity over a period of time. Note that in this strict scientific sense, acceleration can have positive and negative values -- respectively called acceleration and deceleration (or retardation) in common speech -- as well as change of direction. Acceleration is defined technically as "the rate of change of velocity of an object with respect to time" and is given by the equation
An alternative equation is:
After defining his theory of Special Relativity, Albert Einstein realized that forces felt by objects undergoing constant acceleration are indistinguishable from those in a gravitational field, and thus defined General Relativity (which also resolved how gravity's effects could be limited by the speed of light, but that is another story).
A key point of General Relativity is that it solved the "why does only one object feel accelerated?" problem which had plagued philosophers and scientists since Newton's time (and caused Newton to endorse absolute space). Simply put, if you hop in your car and accelerate away from your friend, you could say (given your frame of reference) that it is your friend who is accelerating away from you, although only you feel any force. This is also the basis for the popular Twin paradox, which asks why only one twin ages when moving away from his sibling at near light-speed and then returning, since the aging twin can say that it is the other twin that was moving.
In special relativity, only inertial frames (non-accelerated frames) can be used and are equivalent; general relativity considers all frames, even accelerated ones, to be equivalent. With changing velocity, accelerated objects exist in warped space (as do those that reside in a gravitational field). Therefore, frames of reference must include a description of their local spacetime curvature to qualify as complete.
The rate of change of acceleration is known as jerk or jolt.