Compared to the simple minded raster graphics, which represents an image as a two-dimensional array of "pixels", vector graphics defines images as a collection of geometrical shapes expressed as mathematical equations. This provides some key advantages, such as reduced file sizes and smooth scaling. For example, drawing a circle as a raster graphics image creates hundreds or thousands of pixels, which takes a lot of bytes to store and also makes it difficult to enlarge or move around.
Using vector graphics, a circle is "drawn" simply by specifying the center of the circle and the radius. The rendering software then uses this information to generate the raster images, needed for display or printing, on the fly. A graphics image is thus broken down into as many small geometric components as necessary to produce the desired level of detail.
Inkscape combines vector graphics and raster graphics capabilities. Additionally text can be added to an image in flexible ways, using the fonts installed on your computer. The text can be converted to "paths", making it part of the vector graphics representation. Once the graphic is completed Inkscape allows you save it in standard image formats, such as PNG.
Scalable Vector Graphics (SVG), a royalty-free and vendor-neutral open standard, is a language for describing two-dimensional vector and mixed vector/raster graphics in XML. It is widely adopted and supported by corporations including the mobile phone industry.
A step-by-step example for designing a book cover can be found here.
The graphics software Xara Xtreme provides similar functionalities as Inkscape but uses very efficient implementations.