What is a fraction?
A fraction is a number written in the form 
, where q is nonzero. Any rational number can be written as a fraction. Fractions are usually used to think about "parts of a whole." For example, if someone steals all but one-fifth of your ball of wax, you'll be four-fifths shy of having a whole ball of wax. Gross.
To understand fractions, it may be helpful to think about brownies. Unless you're on a diet. In which case, just mentally replace every occurrence of "brownies" in the following example with the words "veggie squares."
Sample Problem
If we cut a pan of warm, double chocolate caramel brownies (man, those would be some hi-cal veggie squares) into q pieces of equal size and take p of those gooey brownies, the fraction of the pan of brownies we have is 
because we're taking the p from the q. The size of each brownie is 
.
So if we cut a pan of brownies into 4 pieces and take 3 brownies, we'll have 
of the entire pan since each piece is 
of the whole.
Never mind the oddly-shaped pan we used to bake these brownies. We have an old, oddly shaped oven.
Getting back to this fraction of ours, we call the number on top of the line the numerator and the number below the line the denominator.
Numerator comes from the word "numerate," meaning "to number." The numerator tells you how many pieces you have. Denominator comes from the word "denominate," meaning "to give a name to." The denominator gives a name to the pieces, according to their size (for example, "fourths" or "fifths"). Just think, if you had a big litter of nameless puppies, you could numerate and denominate them at the same time.
If the fraction we're looking at is less than 1 (the numerator is less than the denominator), the fraction is called a proper fraction. A fraction that's greater than or equal to 1 is called an improper fraction. Especially if it's using its dessert fork to eat its salad.