Powers and Exponents
Source: ThinkQuest
When dealing with exponents, remember that exponents are a "short cut" to show that a number is to be multiplied by itself a given number of times. For example, x2 is the same as x * x. The number or symbol (variable) that is to be multiplied by itself is called the base (in the example given above, the base is x), and the number or symbol showing how many times it is to be multiplied by itself is called the exponent or power (in the example above, the power is 2).

Video Tutorials
Addition was nice. Multiplication was cooler. In the mood for a new operation that grows numbers even faster? Ever felt like expressing repeated multiplication with less writing? Ever wanted to describe how most things in the universe grow and shrink? Well, exponents are your answer! This tutorial covers everything from basic exponents to negative and fractional ones. It assumes you remember your multiplication, negative numbers and fractions.
Watch the series of Khan Academy The World of Exponents. This series of videos will give you a solid foundation for understanding of exponents. It includes the following topics:

Vocabulary
On the computer, the exponent is typed by using the "superscript" feature. Superscript makes the number or letter a little smaller font size and then raises it up. Sometimes there isn't a "superscript" feature and people use a ^ (also called a carrot) instead.
If we multiply a number times itself we get a “Square” number. These numbers are part of the normal times tables, and form the areas of square shapes. Source: Passy's World of Mathematics

If we multiply a number together three times, then we generate the volume of a cube. These numbers are called “cube numbers” or “Cubics”. Source: Passy's World of Mathematics

Other Tutorials
Here are some other sites that explain how Exponents work:
@HomeTutor  This provides a stepbystep explanation and online exercises to check for understanding.
Math Goodies  This site provides a lot of examples including vocabulary and how it should be read aloud. There are also some online exercises at the bottom.
Math is Fun  This site does a great job explaining how exponents work. You can stop after "Other Way of Writing it" and before "Negative Exponents". We'll learn more about how to work with exponents later.
Online Activities
Here are some fun ways to practice your knowledge of Exponents: