Vector graphics are images that are completely described using mathematical definitions. The image below shows the principle. To the left you see the image itself and to the right you see the actual lines that make up the drawing. An example of a program that uses vectors is Illustrator. Essentially you establish points called vectors that are connected by lines and curves which mimics actual drawing.
Each individual line is made up of either a vast collection of points with lines interconnecting all of them or just a few control points that are connected using so called bezier curves. It is this latter method that generates the best results and that is used by most drawing programs. This technology is referred to as resolution independent, it does not depend on pixels to generate an image. *(for more on this click here to view "how do pixels work?").
* The important thing to remember here is that Illustrator is resolution independent! Since Illustrator does not use pixels to generate the image it can be scaled up or down to any size and the image will maintain its integrity. So doing things like graphics for signs or logos or even a business cards it would make more sense probably to use a program like Illustrator for ease of manipulation. Another way to think of it is that illustrator is for illustrating and creating images from scratch, where as Photoshop is used to manipulate and edit existing photographic images.