With vector graphics, the image is a collection of mathematically defined objects. For example, a circle can be represented by the coordinates of its center point, radius, color, and line thickness.
Objects in a vector image can easily be moved, resized, combined with other objects, etc.
Adobe Illustrator and Microsoft Visio are examples of stand-alone vector drawing programs, and Dabbleboard is a networked vector drawing program with capability for synchronous collaboration and community-contributed image libraries.
A video is worth 1,000 words, and you can see what a vector graphics program is like by watching this video tour of Dabbleboard.
As an historical note, vector drawing preceded raster graphics for economic reasons. The first drawing system was Ivan Sutherland's Sketchpad, which had a great influence on computer graphics and user interface design. (Here is a short video, narrated by Alan Kay, which shows Sketchpad in action). As technology improved, the more familiar and flexible raster graphics became feasible. The Apple Macintosh ushered in the era of raster graphics for the consumer. (See this note on operating system evolution).