Vector graphics are a resolution-independent, scalable format composed of individual objects made up of mathematical calculations. Vector images can be resized easily without loss of quality making them an ideal format for initial design of logos and illustrations that to be used at multiple sizes.
Free Vector Clip Art is a collection of sites with free vector artwork (and some of my own original images too) in .eps, .ai (illustrator), and other vector (and sometimes bitmap) formats. Most are offered with a Creative Commons license that allows for personal and commercial use.
Illustration or drawing software such as Adobe Illustrator and CorelDRAW are popular graphics software for creating and editing vector graphics.
If you have the Flash plug-in, see the Vector definition at About Graphics Software for an online graphic that demonstrates the scalability of vector images.
Although some graphic artists create almost photorealistic vector images, in general vector graphics tend to have a somewhat artificial appearance.