After passing by this question of mine recently, I thought I should give it an answer as I'm now able to do it.
Normal modes is a concept quite beautiful - @Andrew Steane almost answered it for me actually - but basically you have this super complicated movement, or it could be a wave, but the idea is it is something really complex and complicated, and the premise is you can actually recreate this movement using overlays of really simple movements.
In fact, if we're talking for example in a 1-D coupled oscillator, which can generate pretty complex movements to analyse, you can break it down in actually two simple movements, and the complex movement will be a sort of overlap of these two simple movements. These two simple movements are called normal modes.
The idea is exactly the same with vectors, you can represent any vector, no matter how complicated it is, with the basis vectors. Normal modes work as basis vectors (in fact, if they're correctly normalized, they're the same thing) - you can sum them up to recreate the super complicated movement that is happening.
The visualization of adding simple movements that ultimately gives the actual movement is not something easy to imagine, but to convince me that movements can actually be summed up I like to think in two simple movements, for example a 1-D oscillator vibrating with a certain frequency (in a certain direction, sort of speak) and the other simple movement would be the same 1-D oscillator vibrating with the same frequency but with a difference in phase of $\pi$ (in other words, it's vibrating in the opposite direction of the first movement, sort of speak). It makes sense that if we add these two simple movements (with equal amount of contribution of each movement), the overall effect would be that the 1-D oscillator would be in rest, as the simple movements cancel each other out. So it makes sense that we can generate other movements summing other movements.
I hope my answer could help you understand the intuition behind normal modes, something back then it was something hard for me to grasp.