An average is a number that is given to represent a range of numbers. For example, everyone has a different IQ (if you believe in that kind of stuff), but the average IQ is 100. This gives you a basic value to compare any given case to, and it gives you a good guess as to how intelligent the next person you meet on the street is likely to be. There are many different possible ways of creating an average, but the three in common usage are the mean, the median, and the mode.

Mean -- This is what is most often ment by 'average'. It is also known as the mathematical average. To get the mean, you add all the numbers in your sample together and then divide by the number of numbers you had originally.
3, 6, 9 (three numbers)
3+6+9=18
18/3=6
The mean is 6.

Median -- The number in the middle of the set when arranged in numerical order. If you have an even number of numbers in your set, take the center two and get their mean.
1,2,9
2 is the median.

Mode -- The number that occurs the most often.
1,6,7,7,7,121
7 is the mode.

There are others, including the geometric mean, harmonic mean, the root mean square (AKA the quadratic mean), the cubic mean, and the midrange.