Reciprocal
The reciprocal is simply: 1/number
To get the reciprocal of a number, we divide 1 by the number.
Example: the reciprocal of 2 is ½ (a half)
More Examples:
Number  Reciprocal  As a Decimal 

5  \frac{1}{5}  = 0.2 
8  \frac{1}{8}  = 0.125 
1000  \frac{1}{1000}  = 0.001 
Like Turning the Number Upside Down
We can think of a whole number as being "number/1", so the reciprocal is like "flipping it over":

For Fractions, Flip the Whole Fraction Over
Example: the reciprocal of ^{3}/_{4} is ^{4}/_{3}
Read more at Reciprocal of a Fraction.
Flipping a Flip
The reciprocal of a reciprocal takes us back to where we started:
Example:
The reciprocal of 4 is ^{1}/_{4}
The reciprocal of ^{1}/_{4} is 4 (back to 4 again)
It may help to remember that "Reciprocal" comes from the Latin reciprocus meaning returning. Like going to the shops and then returning home again.
But Not Zero
Every number has a reciprocal except 0 (1/0 is undefined)
Multiplying a Number by Its Reciprocal
Let us try multiplying a number by its reciprocal:
How about another:
Try some more yourself! You should always get 1
And that is a way to define Reciprocal:
Reciprocal: what to multiply a value by to get 1
The reciprocal is also called the "Multiplicative Inverse".