Vector art refers to art files that are resolution-independent.
Adobe Illustrator & Macromedia Freehand produce vector art files. Typically only designers, printers, create & use these files. EPS, is an example of vector file. The art is stored as math & coordinates. The files can be very small, and can be scaled up or down without losing image quality.
Raster art refers to art files that are resolution-dependent. Adobe photoshop creates raster files. ALL of the graphics you see on the web are raster files... JPG, GIF, for example. The art is stored as one pixel for one space. IF you scale these images up, you will drastically lose image quality, and the storage size is large.
For our purpose, we trace arcade art with vector tools. Then we share it, so that others can reuse it at any size. Users don't have to worry about DPI, or about how they will get it to the printer (the files are often small enough to email) and they can take bits & pieces and combine them without too much trouble.
A good place to start learning how to vector trace (or "vectorize") art is with Frostillicus's tutorial
http://localarcade.com/4images/details.php?image_id=223&mode=search