is basically a graphical representation of the possible electron positions at various energy levels. The horizontal axis represents distance from the nucleus of the atom. For higher energy levels (increasing $n$), we generally expect an electron within that energy level to be further from the nucleus than an electron at a lower energy level.
The square of the magnitude of the wave function $\psi(x,t)$ (from the Schrödinger equation) gives the probability density associated with finding an electron at a certain position $x$ from the nucleus. This is also sometimes plotted on the potential well (but for the above picture $\psi(x,t)$ is shown because the square of the function should be symmetrical about the central vertical line). We can observe that an electron at a particular energy level (e.g. $n = 1$) can, at one instance, be closer to, or further from, the nucleus than if we were to measure the distance at another instance, and this is based on $\psi(x,t)$ for that particular energy level. An electron at a higher energy level (e.g. $n = 2$) will have an increased "maximum" distance from the nucleus (represented on the potential well diagram), but can sometimes be closer to the nucleus than an electron at a lower energy level, but the probability is again based on the wave functions $\psi(x,t)$ for the two energy levels. More likely than not an electron at a higher energy level will be further away than an electron at a lower energy level.
In reality, the energy levels are split into subshells and we represent 4 different subshells with the letters s, p, d, and f. In a Cartesian coordinate system, we can use 3 different potential well diagrams to represent each spacial dimension. The aomic orbital for any given subshell is a region of space where an electron of that subshell has a 90% chance of occupying at any time. The atomic orbital for the s subshell is spherical (or dome shaped) because the wave function is the same for each spacial dimension, but, for other subshells such as p, the wave function is not the same for every dimension so their atomic orbitals appear to extend in particular directions.
So basically my question is that, based on my explanation, am I right in my understanding, or am I way off? If there is anything that wasn't quite right, please feel free to correct me. I also appreciate additional explanatory comments in general.
Right now, I don't really know much about the Schrödinger equation. As far as I am aware $|\psi(x,t)|^2$ gives the probability density of finding an electron at a certain position, but I don't know how it came about, how to interpret the Schrödinger equation, and how to solve for $\psi(x,t)$. So I plan to do more digging on this. However it would also be great if anyone can provide a dummy explanation to help me understand the Schrödinger equation.