A “raster” is a grid of colored dots. Imagine taking a sheet of graph paper and drawing a picture by coloring in squares with colored pencils, and you’ve got a pretty good idea as to what a raster is. You can do a lot with rasters…what you CAN’T do is enlarge them without taking a quality hit. A “vector” is a computer program. It is a series of commands that tell the output device, “draw an oval. Make it five feet long by two feet high, color it orange and put a black line around it.” It will proceed to do just that, at the highest resolution the output device supports.
For most of you folks, the major difference between vector and raster is going to be that the files can be smaller in vector (although I’ve seen some massive vector files), and small type will look better in vector but you can get nicer blends in raster.