A vector graphic is a graphic based on mathematical objects. Normal Photoshop, jpeg, and gif images are bitmapped images. They have a fixed number of pixels, and the more you enlarge them, then fuzzier the image gets. Try to shrink them, and things bleed together.
Illustrator and postscript files are vector based images. They are stored as mathematical objects. When you resize them, they keep the quality of the original size because instead of adding and removing pixels, you are adding and subtracting to the mathematical formula the object is made of.
Of course you can't display Illustrator and postscript images directly in a web browser, so after resizing, you need to convert to jgep or gif, but the final image is much clearer than resizing a jpeg or gif image. If your original image is a Photoshop, jpeg, or gif, converting to vector can be a pain and time consuming process, but once finished, a vector graphic can be used as a master image for lots of projects.
