Dividing Fractions
Turn the second fraction upside down, then multiply.
There are 3 Simple Steps to Divide Fractions:
Step 1. Turn the second fraction (the one you want to divide by) upsidedown (this is now a reciprocal). 

Step 2. Multiply the first fraction by that reciprocal Step 3. Simplify the fraction (if needed) 
Example:
1  ÷  1 
2  6 
Step 1. Turn the second fraction upsidedown (it becomes a reciprocal):
1  becomes  6 
6  1 
Step 2. Multiply the first fraction by that reciprocal:
(multiply tops ...)
1  ×  6  =  1 × 6  =  6 
2  1  2 × 1  2 
(... multiply bottoms)
Step 3. Simplify the fraction:
6  =  3 
2 
With Pen and Paper
And here is how to do it with a pen and paper (press the play button):
To help you remember:
♫ "Dividing fractions, as easy as pie,
Flip the second fraction, then multiply.
And don't forget to simplify,
Before it's time to say goodbye" ♫
Another way to remember is: "leave me, change me, turn me over" 
A Trick that May Help
Try to rewrite the question the other way around ...
You can rewrite a division question like 20 divided by 5 into "how many 5s in 20?" (=4)
So you can also rewrite ^{1}/_{2} divided by ^{1}/_{6} into "how many ^{1}/_{6}s in ^{1}/_{2}" (=3)
Further Explanation ...
When you divide, you are cutting something into equal shares.
1  ÷  1  is really asking: 
2  6 
How many  1  in  1  ? 
6  2 
Now look at the pizzas below ... how many "1/6th slices" fit into a "1/2 slice"?
How many  in  ?  Answer: 3 
So now you can see why  1  ÷  1  = 3  
2  6 
Another Example:
1  ÷  1 
8  4 
Step 1. Turn the second fraction upsidedown (the reciprocal):
1  becomes  4 
4  1 
Step 2. Multiply the first fraction by that reciprocal:
1  ×  4  =  1 × 4  =  4 
8  1  8 × 1  8 
Step 3. Simplify the fraction:
4  =  1 
8  2 
Fractions and Whole Numbers
What about division with fractions and whole numbers?
Make the whole number a fraction, by putting it over 1.
Example: 5 is also  5 
1 
Then continue as before.
Example:
2  ÷  5 
3 
Make 5 into ^{5}/_{1} :
2  ÷  5 
3  1 
Step 1. Turn the second fraction upsidedown (the reciprocal):
5  becomes  1 
1  5 
Step 2. Multiply the first fraction by that reciprocal:
2  ×  1  =  2 × 1  =  2 
3  5  3 × 5  15 
Step 3. Simplify the fraction:
The fraction is already as simple as it can be.
Answer =  2 
15 
Example:
3  ÷  1 
4 
Make 3 into ^{3}/_{1} :
3  ÷  1 
1  4 
Step 1. Turn the second fraction upsidedown (the reciprocal):
1  becomes  4 
4  1 
Step 2. Multiply the first fraction by that reciprocal:
3  ×  4  =  3 × 4  =  12 
1  1  1 × 1  1 
Step 3. Simplify the fraction:
12  =  12 
1 
Remember the "Trick to Help You" ...
You can rewrite a division question like "20 divided by 5" into "how many 5s in 20" (=4)
So you can also rewrite "3 divided by ¼" into "how many ¼s in 3" (=12)
Why Turn the Fraction Upside Down?
Because dividing is the opposite of multiplying!
A fraction says to:  

But for DIVISION we:
 divide by the top number
 multiply by the bottom number
Example: dividing by ^{5}/_{2} is the same as multiplying by ^{2}/_{5}
So instead of dividing by a fraction, it is easier to turn that fraction upside down, then do a multiply.