Cross Multiply
To cross multiply is to go from this:  \frac{8}{12} = \frac{2}{3} 

To this:  8 × 3 = 12 × 2 
How Does it Work?
Multiplying the top and bottom of a fraction by the same amount doesn't change its value.
Step 1: Multiply the top and bottom of the first fraction by the bottom number of the second fraction.
\frac{8 × 3}{12 × 3} = \frac{2}{3}
Step 2: Multiply the top and bottom of the second fraction by the bottom number the first fraction had.
\frac{8 × 3}{12 × 3} = \frac{2 × 12}{3 × 12}
And Magic! The bottom of both fractions is now 12 × 3
Step 3: We can get rid of the 12 × 3 (as we are dividing both sides by the same amount) and the equation is still true:
8 × 3 = 12 × 2
Job Done!
In practice, though, it is easier to skip the steps and go straight to the "crossmultiplied" form.
Using Variables
The general case, using variables instead of numbers, is:
To cross multiply is to go from this:  \frac{a}{b} = \frac{c}{d} 

To this:  ad = bc  
To remember think cross (x) multiply:
Cross multiplication can help speed up a solution. Like in this example:
Example: Find "x" here:
\frac{x}{8} = \frac{2}{x}  
Let's cross multiply:  x^{2} = 8 × 2  
Calculate:  x^{2} = 16  
And solve:  x = 4 or −4 
Check: Does \frac{4}{8} = \frac{2}{4} and \frac{−4}{8} = \frac{2}{−4} ? Yes!
Terminology
I said "top" and "bottom" of the fractions ... but the correct words are numerator and denominator, OK? (I just wanted to keep it simple.)
Caution: Zero
Be careful, though!
We cannot use it when a denominator ("b" and "d" above) is zero, as dividing by zero is "illegal".