Acceleration is defined as change in the velocity of an object. Thus, any object which has been set in motion from rest, or stopped from moving, or which has changed direction of motion from a straight line, has undergone some acceleration.
The mathematical dimensions of acceleration are dV/dt, or in other words, change of velocity with respect to time. This is equivalent to (dx/dt)/dt or dx/dt² and so acceleration involves itself, only distance and time.
The acceleration of any isolated object is evidence that some external force has acted upon it (as expounded in the physical laws of Isaac Newton). This accelerating force can be described mathematically by the equation F = mA (force = mass x acceleration), which can be rearranged to A = F/m. Therefore, acceleration is directly proportional to the force acting on an object, and inveresely proportional to the objects mass, which correlates with everyday experience; the greater the push, the faster an object is accelerated, and the heavier the object, the more difficult it is to accelerate.
The observation of accelerations is of fundamental importance in experimental physics. It is through these observations that otherwise invisible forces can be detected and quantified. For example, it was through the observation of the acceleration of an apple downward from a tree that Sir Isaac Newton was led to the hypothesis of universal gravitation. And, it is possible through such observations to quantify not only the force involved, but also the mass of an object (if the force is known) - this is how the mass of subatomic particles can be measured, for example.
As the relative velocity of an object approaches the speed of light, the mathematical description involved in acceleration becomes more complicated; more and more of the kinetic energy of an object becomes manifested as an increase in its mass, making acceleration progressively more difficult. This is of no consequence in observations made under 'ordinary' conditions, however.