Inkscape is an open-source vector graphics editor similar to Adobe Illustrator, Corel Draw, Freehand, or Xara X. What sets Inkscape apart is its use of Scalable Vector Graphics (SVG), an open XML-based W3C standard, as the native format.
In contrast to raster (bitmap) graphics editors such as Photoshop or Gimp, Inkscape stores its graphics in a vector format. Vector graphics is a resolution-independent description of the actual shapes and objects that you see in the image. A rasterization engine uses this information to determine how to plot each line and curve at any resolution or zoom level.
Contrast that to bitmap (raster) graphics which is always bound to a specific resolution and stores an image as a grid of pixels.
Vector graphics are a complement, rather than an alternative, to bitmap graphics. Each has its own purpose and are useful for different kinds of things. Raster graphics tend to be better for photographs and some kinds of artistic drawings, whereas vectors are more suitable for design compositions, logos, images with text, technical illustrations, etc.
Note that Inkscape can import and display bitmap images, too. An imported bitmap becomes yet another object in your vector graphics, and you can do with it everything you can do to other kinds of objects (move, transform, clip, etc.)