Raster graphics are made of pixel information. The best example of this is a picture you take with your digital camera. Raster graphics are because of their nature are only as good as the pixel ratio to size the are printed at. Meaning that the bigger you print it the higher the resolution of the original needs to be to maintain quality. This is why as technology has evolved today's digital cameras print much better pictures than the early ones did. They are capable of capturing much more resolution aka detail. Raster images only need be 72 to 96 dpi to look good on your monitor. To look good in print it is typically a minimum of 300 dpi.
Vector art most commonly is illustration are or things like typefaces aka fonts. Most company logos are good examples of vector line art. They contain pixels instead they are line or as they call it outline based. Meaning that the edge is a line and as such vector art is infinitely scale-able without quality loss.
You will see this explained many different ways depending on what you are doing with it in the end, be it print, video or the web. This is about as simple as I can explain it. Hope it helps.