Simply talking, we can say that White Balance is that thing you have to properly set in the camera to have the real colors! If you want an extensive technical explanation, go to the wiki.
So white balance is just a method to find the right color balance. That means “understand when white is white” and from that point, understand the real color of things in the image. The automatic white balance (AWB) in almost every digital camera, does not match 100% the colors and sometimes we have to specify what kind of light we are exposing the picture.
What kind of light? Light is just light! Different light sources produce different temperature lights. For example, the light in a cloudy day is totally different from the direct light of a sunny day or from the light of an electric neon tube. The colors render differently and our eyes and brain adjust automatically to that light to show us colors as much close as possible to the original.
An image will clarify this better.