A vector image is an image format in which the data is stored as coordinates, lines, curves, gradients etc rather than pixels. The final rendering of the image is done on the machine viewing the file, so it can be rendered at any size without being pixelised.
A common use of vector images is for icons. Distributing icons at several pre-rasterised sizes is much less efficient and flexable than as a single file that can be rasterised to any size. Distributing as raster images will also give worse looking results if the user doesn't have icons size set to one of the distributed sizes.