A vector, in terms of digital artwork, refers to any artwork drawn with bezier
curves. Bezier curves define an image in terms of mathmatical formula that define the shape of a line. Because
vectors are saved as a function of these formulae, they can be scaled up and down without loss in quality. This is in
comparision to bitmap images which are defined by a grid, each pixel in the grid
having a color assigned to it.
To create vectors you need a program capable of it. Inkscape and Sodipodi are two free/shareware programs; Adobe
Illustrator, Macromedia Flash and Freehand, and Corel Xara are professional software that specialize in vector
illustration. Some other art programs like Photoshop mimic vector illustration but are not usually considered
"true" vector programs.
In a vector trace, people generally just trace over an existing image with vectors. This basically converts a bitmap
image into a vector image, and allows the user to both clean the image (since the image is entirely redrawn) and create
an easily scalable image (especially from small sources such as screencaps.)
For more information, try searching for "bezier curve" to learn more about the mathematics of vectors.