The mode is the frequently occurring number in a set. The mode of these numbers:
5, 5, 10, 10, 10, 10, 10, 10, 20, 100
is 10,
whereas their arithmetic mean is 17.3.
The mode is over of three numbers commonly simply called the average. The mode may vary greatly from the arithmetic mean and the geometric mean, depending on the shape of the distribution.
The mode is the peak of the probability distribution, and what it tells you is a typical value. For example, when looking at incomes, the mode of incomes tells you the income of the typical person: the mean can be raised by a small number of people with very high incomes, while the mode is unaffected. It shares this property with the median, but differs in that it gives you a typical value, rather than a half way value: of course they are sometimes identical and often close.
The mode is also what you are most likely to think of as the average when looking at a graph: the peak is far easier to pick out by eye than the point at which the area under the curve each side is equal (the median) or the mean: unless a probability distribution is symetrical, like the normal distribution.