The principal quantum number is the label we use for unique energy eigenvalues for a given Hamiltonian. Based on your question, I believe I need to step back and explain what I mean by this.
The orbitals that you learn about in chemistry are technically the orbitals for a single electron and a nucleus (a "Hydrogenic" atom). When you have multiple electrons, it gets far more complex, as the energy of one electron depends on where all of the other electrons are, and the nucleus can be screened by the other electrons. So it’s not accurate for multielectron systems, but it’s close enough to get the rough idea across, and solving the entire problem would be too complicated, so it is good enough.
Ok, fine, so what is the problem that we are solving with the hydrogen atom then? Well with any quantum system, solving the full time dependent Schrödinger Equation is pretty hard, and for all but a few systems (infinite well, harmonic oscillator, step function, linear, hydrogen atom) you can’t even solve the time independent equation at all by hand. I’m not sure how much you know, but the Schrödinger equation describes the motion of non-relativistic particles, that’s why I’m referring to it.
It turns out that if the Hamiltonian is time independent (which is the case for hydrogen), that the solutions are of the form $\Psi(\vec{r},t) = \sum_n c_n\psi_n(\vec{r}) e^{\frac{E_n}{i\hbar}t}$, i.e. you can separate the time dependence. This means that if you can find the $\psi_n(\vec{r})$ (the so-called "energy eigenfunctions"), then you can expand any initial state in terms of those functions. Then to see how the wave function changes in time, you just plug in the time into that exponential.
The idea is that we find states whose amplitude (where we interpret the square of the amplitude as the probability density) is time independent, i.e. constant in time, (however phase changes at a rate proportional to $E_n$, the energy eigenvalue). This is what solving the time independent Schrödinger equation is.
When you find the set of energy eigenfunctions, it is possible that multiple functions share the same eigenvalue. This is the case for hydrogenic atoms, where different $l$ and $m$, which are the eigenvalues that correspond to $\vec{L}^2$ and $L_z$, the magnitude of the angular momentum squared, and the z component of the angular momentum respectively. These operators commute with the Hamiltonian, which in this context means that their eigenstates are also time independent.
So in short, the principal quantum number $n$ is a label that tells you what the energy of the state is, and the others are related to orbital angular momentum. The reason that different $l$ values for the same orbital give you different $n$ is that some of the energy is rotational.
The solutions for the hydrogen atom can be separated into a radial and angular part, $\psi_{n,l,m} = R_{n,l}(r)Y_{l,m}(\theta, \phi)$, where $Y_{l,m}$ are the spherical harmonics and $R_{n,l}$ are the radial wavefunctions https://quantummechanics.ucsd.edu/ph130a/130_notes/node237.html#derive:Hradial
Note that the radial wavefunctions are determined by BOTH $n$ and $l$. And the higher $l$ is, the fewer antinodes there are in the radial component. However, a general trend in quantum is that the more antinodes there are, the higher the energy is. So where did the antinodes go? They went to the spherical harmonics. If you look at a table of spherical harmonics, you see that the higher $l$ is, the more antinodes there are, while $m$ just changes where the antinodes are. https://en.m.wikipedia.org/wiki/Spherical_harmonics#/media/File%3ARotating_spherical_harmonics.gif
Something to note, since the electron is moving rather fast around the nucleus, the electric field becomes Lorentz transformed a slight amount into a magnetic field which couples to the spin of the electron. This means that spin up has a slightly different energy from spin down. This breaks the symmetry of the Hamiltonian with respect to spin, and as a result “lifts the degeneracy” of the states. This is called hyperfine splitting, and causes spectral lines to split, which you can see if you look very closely at the spectrum.
