Instead of trying to keep track of the millions of tiny pixels in a raster image, vector images, or line art, keep track of points and the equations for the lines that connect them. Generally speaking, vector images are made up of paths or line art that can scale to any size because they are made mathematically.
One of the greatest things about vector images is that you can re-size them infinitely larger or smaller, and they will still print out just as clearly, with no increase (or decrease) in file size. Because the computer does not need to process hundreds or thousands of separate pixels, the file size of vector images is quite small compared to raster images. This allows your computer to process these files much more quickly.
So, what types of graphics would typically be vector? Well, almost all computer font files are based on vector images of the letters - that's why it's possible to scale them WAY up or WAY down and still have the letters be clear. All Microsoft Office clipart (discontinued in Office 2013, but still searchable in Bing) uses vector art and most charts, shapes and graphs produced by Office are vector-based. Vector illustrations are great for logos, illustrations/artwork, animations, and text.
Common Vector File Types: EPS, SVG