Graphics come in two types, raster and vector. Photoshop, Paintshop Pro, and Photopaint are all examples of raster programs. Raster images are described by an array of pixels that each have a given color and intensity value.
Vector images are described by mathematical lines, curves, and fills. Adobe Illustrator, Corel Draw, and Macromedia Freehand are examples of vector programs.
Both raster and vector images have their advantages and disadvantages. Raster images look more realistic, but vector images print more easily and can be infinitely scaled up or down with no loss of quality.
When you 'vectorize' a raster image, you translate it into a system of lines and curves. There are programs specifically for this, such as Adobe Streamline. Also, many vector programs have the function built in. The opposite, called 'rendering', takes the mathematical description of a vector graphic and converts it into discrete pixels.