Once upon a time, in ancient China, there was a terrible flood.  The people offered sacrifices to the god of the Lo river so that the river would stop flooding.  Out of the receding flood waters emerged a turtle with strange markings on its shell.  The people realized that the markings were numbers arranged in a square.  More amazing, when they added the numbers in any row, column or diagonal, they always equaled fifteen.

This is the story of the first magic square, called the Lo Shu square.  As the story tells it, the markings were drawn as dots arranged to represent the numbers from one to nine.  My recreation is here on the right:

As you can see, if we add up the left column (4+3+8), or the middle column (9+5+1), or the bottom row (8+1+6), or a diagonal (4+5+6), we get 15.  In fact, any column, row, or diagonal we add, we get 15.  

Magic squares don't always have to be 3x3 squares--in fact, the 3x3 square is in some ways not a very interesting one.  Every 3x3 magic square with numbers one through nine will be some rotation or reflection of this one.  Magic squares can be much bigger, or they can use numbers that aren't consecutive.  The Wikipedia article has a lot of great examples.  

These squares are very fun to play around with, and anyone who can add can find patterns.  You can design and create your own magic square here.