Second Derivative
(Read about derivatives first if you don't already know what they are!)
A derivative basically gives you the slope of a function at any point.
The "Second Derivative" is the derivative of the derivative of a function. So:
 Find the derivative of a function
 Then take the derivative of that
A derivative is often shown with a little tick mark: f'(x)
The second derivative is shown with two tick marks like this: f''(x)
Example: f(x) = x^{3}
 Its derivative is f'(x) = 3x^{2}
 The derivative of 3x^{2} is 6x, so the second derivative of f(x) is:
f''(x) = 6x
A derivative can also be shown as:  dy  , and the second derivative shown as:  d^{2}y 
dx  dx^{2} 
Example: (continued)
The previous example could be written like this:
y = x^{3}
dy  = 3x^{2} 
dx 
d^{2}y  = 6x 
dx^{2} 
Distance, Speed and Acceleration
A common real world example of this is distance, speed and acceleration:
Example: A bike race!
You are cruising along in a bike race, going a steady 10 m every second.
Distance: is how far you have moved along your path. It is common to use s for distance (from the Latin "spatium").
So let us use:
 distance (in meters): s
 time (in seconds): t
Speed: is how much your distance s changes over time t ...
... and is actually the first derivative of distance with respect to time: \frac{ds}{dt}
And we know you are doing 10 m per second, so \frac{ds}{dt} = 10 m/s
Acceleration: Now you start cycling faster! You increase your speed to 14 m every second over the next 2 seconds.
When you are accelerating your speed is changing over time.
So \frac{ds}{dt } is changing over time!
We could write it like this: 


dt 
But it is usually written \frac{d^{2}s}{dt^{2}}
Your speed increases by 4 m/s over 2 seconds, so \frac{d^{2}s}{dt^{2}} = \frac{4}{2} = 2 m/s^{2}
Your speed changes by 2 meters per second per second.
And yes, "per second" is used twice!
It can be thought of as (m/s)/s but is usually written m/s^{2}
(Note: in the real world your speed and acceleration changes moment to moment, but here we assume you can hold a constant speed or constant acceleration.)
So:
Example Measurement 

Distance:  s  100 m 
First Derivative is Speed:  \frac{ds}{dt}  10 m/s 
Second Derivative is Acceleration:  \frac{d^{2}s}{dt^{2}}  2 m/s^{2} 
And the third derivative (how acceleration changes over time) is called "Jolt" ... !
Play With It
Here you can see the derivative f'(x) and the second derivative f''(x) of some common functions:
Notice how the slope of each of those functions is the derivative plotted below it.