Vectors are essentially coordinates that describe a point or line. Take (2,4). Picture a graph, with x being 2 and y being 4. That point is the vector. There are also 3D vectors like (2,4,4), which is the same except for the inclusion of a third dimension.
Vectors are very important in graphics programming. They can do many things, but most importantly they tell the program where exactly the screen, and therefore the player, is facing. For example, an FPS needs to be able to determine if the player is facing an enemy, so that it knows to draw that enemy onto the screen. Everything that the player is not facing can be disregarded, graphically at least. If computers had to draw everything the player could be facing, the program would run much too slow.