Vector graphics use pure mathematics to draw shapes on a computer monitor, or on an output device such as a printer.
For example, consider a circle of radius r. The only pieces of information a computer needs in order to draw this circle are a) the radius r, b) the location of the center point of the circle, c) the circle color, d) line thickness, e: line style, and f) fill in or just draw an outline of the circle.
There are two major advantages to this style of drawing over raster graphics. First, this minimal amount of information translates to a much smaller file size. Second, the size of items, similar to our circle radius r, above, can be considered relative to the size of the containing rectangle. This means that vector graphics can scale (be resized) without any loss of quality. For example, if we have a bounding box that is 256x256 pixels, and a circle of radius 128 located at x,y coordinates (128, 128) we can scale the whole thing by multiplying all of these numbers by the same factor, say x4, and end up with a bounding box that is 1024x1024 and a circle that has a radius of 512 and a center point of (512,512). The circle drawn at this new size loses no quality over the smaller 128 radius circle.
Vector graphics are ideal for simple or composite shapes that need to scale, or do not need to achieve photo-realism.
In 3D computer graphics, vectorized surface representations are common. At the low-end, simple meshes of polygons are used to represent geometric detail in applications where interactive frame-rates or simplicity are important. At the high-end, where one is willing to trade-off higher rendering times for increased image quality, smooth surface representations such as Bezier patches, NURBS or Subdivision surfaces are used.