Vector graphics differ from bitmap or raster graphics in that they are not made up of a grid of pixels. They rely on computer algorithms to define paths and shapes. Vector graphics are rarely used in web design, but are very common (required) in graphic design. A logo, for example, would be created in vector format. From there it is exported to bitmap/raster format for use on a web site.
The main advantage is that vector graphics are fully scalable. You can print your vector logo the size of Texas without any pixelation along the edges.