Short answers, pulled from the story.
Acceleration is the rate of change of velocity, covering changes in both speed and direction of motion. It is a vector quantity, meaning it has both magnitude and direction. The SI unit for acceleration is the metre per second squared (m/s2).
Tangential acceleration is the component of acceleration aligned with the direction of motion, changing an object's speed. Centripetal acceleration is the component directed toward the centre of curvature, changing the object's direction. In uniform circular motion, all acceleration is centripetal; there is no tangential component.
Newton's second law states that the net acceleration of a body is proportional to the net force acting on it and inversely proportional to the body's mass. The direction of the net acceleration matches the direction of the net force.
Albert Einstein's equivalence principle states that gravity and inertial acceleration are locally indistinguishable. Without knowing the state of motion of an object, it is impossible to tell whether an observed force results from gravity or from acceleration.
As speeds approach the speed of light, the acceleration produced by a given force decreases, becoming infinitesimally small as light speed is approached. An object with mass can approach light speed asymptotically but can never actually reach it, as described by special relativity.
Average acceleration is the change in velocity divided by the elapsed time over a period and is the only form of acceleration directly measurable without an empirical law. Instantaneous acceleration is the limit of average acceleration over an infinitesimally small interval, expressed mathematically as the derivative of the velocity vector with respect to time.