When the velocity of an object changes it is said to be accelerating. Acceleration is the rate of change of velocity with time.
In everyday English, the word acceleration is often used to describe a state of increasing speed. For many Americans, their only experience with acceleration comes from car ads. When a commercial shouts "zero to sixty in six point seven seconds" what they're saying here is that this particular car takes 6.7 s to reach a speed of 60 mph starting from a complete stop. This example illustrates acceleration as it is commonly understood, but acceleration in physics is much more than just increasing speed.
Any change in the velocity of an object results in an acceleration: increasing speed (what people usually mean when they say acceleration), decreasing speed (also called deceleration or retardation), or changing direction. Yes, that's right, a change in the direction of motion results in an acceleration even if the moving object neither sped up nor slowed down. That's because acceleration depends on the change in velocity and velocity is a vector quantity — one with both magnitude and direction. Thus, a falling apple accelerates, a car stopping at a traffic light accelerates, and an orbiting planet accelerates. Acceleration occurs anytime an object's speed increases or decreases, or it changes direction.
Much like velocity, there are two kinds of acceleration: average and instantaneous. Average acceleration is determined over a "long" time interval. The word long in this context means finite — something with a beginning and an end. The velocity at the beginning of this interval is called the initial velocity, represented by the symbol v_{0} (vee nought), and the velocity at the end is called the final velocity, represented by the symbol v (vee). Average acceleration is a quantity calculated from two velocity measurements.

In contrast, instantaneous acceleration is measured over a "short" time interval. The word short in this context means infinitely small or infinitesimal — having no duration or extent whatsoever. It's a mathematical ideal that can can only be realized as a limit. The limit of a rate as the denominator approaches zero is called a derivative. Instantaneous acceleration is then the limit of average acceleration as the time interval approaches zero — or alternatively, acceleration is the derivative of velocity.

Acceleration is the derivative of velocity with time, but velocity is itself the derivative of displacement with time. The derivative is a mathematical operation that can be applied multiple times to a pair of changing quantities. Doing it once gives you a first derivative. Doing it twice (the derivative of a derivative) gives you a second derivative. That makes acceleration the first derivative of velocity with time and the second derivative of displacement with time.
a =  dv  =  d  dr  =  d^{2}r  
dt  dt  dt  dt^{2} 
A word about notation. In formal mathematical writing, vectors are written in boldface. Scalars and the magnitudes of vectors are written in italics. Numbers, measurements, and units are written in roman (not italic, not bold, not oblique — ordinary text). For example…
a = 9.8 m/s^{2}, θ = −90°  or  a = 9.8 m/s^{2} at −90° 
(Design note: I think Greek letters don't look good on the screen when italicized so I have decided to ignore this rule for Greek letters until good looking Unicode fonts are the norm on the web.)
Here are some sample accelerations to end this section.