Conjugate
The conjugate is where you change the sign in the middle of two terms like this:
It is only used in expressions with two terms (called "binomials")
Other examples:
Expression 
Its Conjugate 
x^{2}  3 
x^{2} + 3 
a + b 
a  b 
a  b^{3} 
a + b^{3} 
Examples of Use
The conjugate can be very useful because ...
... when you multiply something by its conjugate you get squares like this:
How does that help?
It can help you move a square root from the bottom of a fraction to the top (or vice versa). And I will show you how.

Note: The bottom of a fraction is called the denominator, and many square roots are irrational, so this is called "Rationalizing the Denominator" 
Example: here is a fraction with an "irrational denominator":
How can we move the square root of 2 to the top?
Answer! Multiply both top and bottom by the conjugate (this will not change the value of the fraction), like this:
(Did you see how the denominator became "a^{2}b^{2}"?)
There is another example on the page Evaluating Limits where I move a square root from the top to the bottom.
So try to remember this little trick, it may help you solve an equation one day!
