E- in Box Concept

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Dr. Lavelle went over the concept of how the bound electron can be in different energy states, but it's still bound within the box, with different curves within the box.

I am still a bit confused on this concept.
What does this box represent?
Is this about how electrons are quantized so they can never go "out of the box", meaning "in between" quantized states?
How does this concept relate directly to electrons?

If someone could simplify/summarize this concept for me that would be greatly appreciated.

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Don't worry! It can be confusing at first! We can break it into simpler concepts. The key thing to know is that the electrons can't have any amount of energy. They can jump between energy levels, but the "box" shows that they are bound to a specific energy level in the atom. You were correct: it basically demonstrates the quantization of energy existing in one confined space. If you were to visualize a box (a one-dimensional one), you can imagine that lines of the box representing the limit or boundary that a certain particle cannot touch/pass through. Since the energy of a particle is quantized and only has discrete energy levels, we can also use the size of the box to determine how high the energy level is. (Smaller boxes = higher energy, Bigger boxes = lower energy). By looking at the wave function, we can also see, by probability, where a particle may be positioned within that box. i hope that helps a little bit!