Here's a really helpful website that explains it quite good:

http://www.mathsisfun.com/positive-negative-integers.html

To subtract a negative number, make the number positive and add it instead. So 7 minus negative 8 becomes 7 plus 8.

With any number when combining, you add or subtract based on the symbol behind it. Example:

                  5 + 5 = 10  or 5 - 5 = 0

The positive sign and the negative sign change the equation into addition and subtraction, or you can look at it as if you are adding a positive or adding a negative.

Subtraction and adding a negative are the same.  The  ( - ) symbol means subtraction, but subtracting a poisitve has the same effect if it was a negative number being added. Example:

 5 - 5 = 0 is subtracting positive five, whereas:

 5 - 5 = 0 is adding a negative. There is not a difference.

Now try subtracting a negative.

 5 - (-5) = 10

This equation just means that when a negative is subtracted, since that is the same as the flip, adding a positive, they are the same.

 Again, 

5 - 5 = 0

5 + 5 = 10

5 - (-5)= 10