Raster images/graphics are determined by pixels. Raster images can be comprised of millions of colors.
If you zoom in very close you will see individual blocks of color that when printed out at the correct size will result in an image (picture/photo). As in the image below.
Vector images on the other hand are comprised of paths/lines. Vector images must have a start and finish point. But they don't have to be straight, they can be curved. They are made up of editable paths which can be manipulated. In the image below you can see the nodes (the square dots) around the edge of the image.
Vector images can be reshaped and resized without losing degradation, which is why it is the preferred method for making clip art and logos. Vector images do not lend themselves to photographic imagery very well.
One more thing vector images can be converted to raster (and usually are for printing); but raster cannot be converted to vector. All of the graphic programs I have worked with all you to apply text as a selection, floating or vector. I nearly always choose vector so that I can resize or reshape without having to delete and redoing the layer.
This is a very basic explanation of raster and vector. To get a more comprehensive answer, you can do a search for them. There are a lot of sites out there that can explain it a lot better than I.