Imagine a chain made of a line of springs with some mass $m$ located between each spring and the next.
Now suppose you wobble this system in some complicated wobbly motion (not a normal mode). All the masses will move up and down by different amounts and in some complicated motion. If you could hear the sound waves produced by the motion then it would sound like noise.
Now suppose instead that you carefully arrange that one of the masses goes up and down in a strictly periodic way: just one frequency, just one 'note' if you could hear it. The adjacent masses will pick up this motion too, and then the ones adjacent to those, etc. The motion throughout the system will stay complicated for a while, but eventually it may settle into a pattern where all the masses are going up and down at the same frequency. They don't have to be in phase: some may be going up while others go down, but this phase difference will be constant in time. Also they don't have to have the same amplitude: some may go up and down more than others. If you looked at the system it would like as if it had a standing wave, with all the motions having exactly the same frequency, repeating over and over.
This sort of motion is called a 'mode of oscillation' or just 'mode' for short. If, in the absence of damping, you do not need to provide an external force because the system just carries on oscillating at a single frequency throughout, then it is called a 'normal mode'.
For any given set of springs and masses, the normal mode motion can only take place at one of a discrete set of frequencies (this is what the eigenvalues tell you). And each normal mode has its own characteristic shape of displacements along the system (this is what the eigenvectors tell you).
I talked about springs and masses, but the concept is more general, and has quite a widespread application in physics, pretty much whenever something is more complicated than a single particle and can undergo some sort of oscillatory motion.