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 (v) and the velocity at the end is called the final velocity (v_{0}) [v nought]. Average acceleration is a quantity calculated from 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.)
Calculating acceleration involves dividing velocity by time — or in terms of units, dividing meters per second [m/s] by second [s]. Dividing distance by time twice is the same as dividing distance by the square of time. Thus the SI unit of acceleration is the meter per second squared.
⎡ ⎣ 
m  =  m/s  =  m  1  ⎤ ⎦ 

s^{2}  s  s  s 
Another frequently used unit is the acceleration due to gravity — g. Since we are all familiar with the effects of gravity on ourselves and the objects around us it makes for a convenient standard for comparing accelerations. Everything feels normal at 1 g, twice as heavy at 2 g, and weightless at 0 g. This unit has a precisely defined value of 9.80665 m/s^{2}, but for everyday use 9.8 m/s^{2} is sufficient, and 10 m/s^{2} is convenient for quick estimates.
The unit called acceleration due to gravity (represented by a roman g) is not the same as the natural phenomena called acceleration due to gravity (represented by an italic g). The former has a defined value whereas the latter has to be measured. (More on this later.)
Although the term "g force" is often used, the g is a measure of acceleration, not force. (More on forces later.) Of particular concern to humans are the physiological effects of acceleration. To put things in perspective, all values are stated in g.
Here are some sample accelerations to end this section.
a (m/s^{2})  event 

5 × 10^{−14}  smallest acceleration in a scientific experiment 
2 × 10^{−10}  galactic acceleration at the sun 
9 × 10^{−10}  anomalous acceleration of pioneer spacecraft 
0.5  elevator, hydraulic 
0.6  free fall acceleration on pluto 
1  elevator, cable 
1.6  free fall acceleration on the moon 
8.8  International Space Station 
3.7  free fall acceleration on mars 
9.8  free fall acceleration on earth 
10–40  manned rocket at launch 
20  space shuttle, peak 
24.8  free fall acceleration on jupiter 
20–50  roller coaster 
80  limit of sustained human tolerance 
0–150  human training centrifuge 
100–200  ejection seat 
270  free fall acceleration on the sun 
600  airbags automatically deploy 
10^{4}–10^{6}  medical centrifuge 
10^{6}  bullet in the barrel of a gun 
10^{6}  free fall acceleration on a white dwarf star 
10^{12}  free fall acceleration on a neutron star 
event  typical car  sports car  F1 race car  large truck 

starting  0.3–0.5  0.5–0.9  1.7  < 0.2 
braking  0.8–1.0  1.0–1.3  2  ~ 0.6 
cornering  0.7–0.9  0.9–1.0  3  ?? 
a (g)  event 

2.9  sneeze 
3.5  cough 
3.6  crowd jostle 
4.1  slap on back 
8.1  hop off step 
10.1  plop down in chair 
60  chest acceleration during car crash at 48 km/h with airbag, design limit 
70–100  crash that killed Diana, Princess of Wales, 1997 
150–200  head acceleration limit during bicycle crash with helmet 