What Is A Vector?
A Vector format illustration is created by plotting points along straights, curves and corners to create a desired shape (see example to the right).
These shapes can then be overlaid, coloured in, and moved about at will.

They differ from Raster images, which is the format used in Photoshop, in that they can be resized up or down without any loss of image quality, from postage stamp to billboard.

They are also much easier to alter at a later date, if say a drawing is revised, than a version in say GIF, or JPG.

You can find a more detailed Wikipedia explanation here
Vector Example