The White Balance determineshow accurately the colours in images look, for example it determines how “hot” or “cold” an image feels. Setting your white balance accurately will prevent against any harsh or nasty colours which makes images look unnatural. On a camera there's also an option for “auto white-balance” (AWB) which is suitable for when shooting indoors in tungsten lighting. Grey cards are used to determine which white balance should be used by your camera. The reason a grey is used over white is because, if you over-expose any colour enough, it will come out as white. White balance is about colour, not brightness.
This white balance diagram shows what white balance can do to a picture. If white balance is set correctly then it can change the tone of a picture, for example the "incandescent" tone illustrates a cold, blue, icy photo which makes the person feel like the picture can mean a lot compared to the "shade" tone which suggests a natural lighting.