A raster image is an array of tiny squares with a given uniform color. The pixels forming a regular grid, only the colors are given, coordinates are implicit.
A vector image is a set of more complex shapes such as polygons, circles and Bezier curves, filled with a solid color (sometimes gradients or textures). The shapes are described by a number of parameters and/or control points.
You understand that the raster representation is only an approximation of the desired circle, using intermediate shades of gray to lessen the staircase effect on the outline. The vector representation truly describes a circle.
In both cases, the graphics card needs to render the images. Processing of the raster is much easier due to the simple nature of the squares.
In the zoomed-in views, notice that the outline of the circle on the left are neat; but the outlines of the pixels on the right are neat too (you see nice squares) !
Also note that in both cases the graphics card convertes the images to a raster having the resolution of the screen (the screen IS a raster image of fixed resolution).