Wireless networks work using radio waves instead of wires to transmit data between computers. That's the simple version. If you're curious to know what's going on in more detail, then it's all explained in this article.
Ones and Zeros.
I'm sure you know that computers transmit data digitally, using binary: ones and zeros. This is a way of communicating that translates very well to radio waves, since the computer can transmit ones and zeros as different kinds of beep. These beeps are so fast that they're outside a human's hearing range -- radio waves that you can't hear are, in fact, all around you all the time. That doesn't stop a computer from using them, though.
The way it works is a lot like Morse code. You probably already know that Morse code is a way of representing the alphabet so that it can be transmitted over radio using a dot (short beep) and a dash (long dash). It was used manually for years, and became a great way of getting information from one place to another with the invention of the telegraph. More importantly for this example, though, it is a binary system, just like a computer's ones and zeros.
You might think of wireless networking, then, as being like Morse code for computers. You plug a combined radio receiver and transmitter in, and the computer is able to send out its equivalent of dots and dashes (bits, in computer-speak) to get your data from one place to another.
All About Frequencies.
You might wonder, though, how the computer could possibly transmit enough bits to send and receive data at the speed it does. After all, there must be a limit on how much can be sent in a second before it just becomes useless nonsense, right? Well, yes, but the key to wireless networking is that it gets around this problem.
First of all, wireless transmissions are sent at very high frequencies, meaning that more data can be sent per second. Most wireless connections use a frequency of 2.4 gigahertz (2.4 billion cycles per second) -- a similar frequency to mobile phones and microwave ovens. As you might know, though, a frequency this high means that the wavelength must be very short, which is why wireless networking only works over a limited area.