You can get better answers by googling for it, but I can try to explain in layman terms..
Vector graphics are created by applications like Inkscape, Adobe illustrator, Freehand and Flash. As opposed to Raster graphics (like the many beautiful JPEGs u see in Clubsnap) which are made up of pixels, vector images are made up of mathematical data (vectors). A rendering engine (e.g flash player, adobe acrobat, etc) will used this data to display them as graphics proper.
Vector editing is preferred in design and illustration because they are easier to manipulate and scale.