What do atomic orbitals represent in quantum mechanics? I am learning the basics of quantum mechanics and am familiar with the Schrödinger equation and its solution, but I was confused about what the familiar atomic orbital shapes represent?
Do they represent nothing physical and are just plots of the wavefunction in 3D polar co-ordinates? Or do they represent the region where probability of finding an electron is $90\%$? Or something else?
Levine 7th ed. states that

An atomic orbital is just the wavefunction of the electron

Wikipedia instead states that

In atomic theory and quantum mechanics, an atomic orbital is a mathematical function describing the location and wave-like behavior of an electron in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus. The term atomic orbital may also refer to the physical region or space where the electron can be calculated to be present, as predicted by the particular mathematical form of the orbital

 A: Let me split up your sources into Levine

An atomic orbital is just the wavefunction of the electron

as well as Wikipedia part 1

In atomic theory and quantum mechanics, an atomic orbital is a mathematical function describing the location and wave-like behavior of an electron in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus.

and Wikipedia part 2.

The term atomic orbital may also refer to the physical region or space where the electron can be calculated to be present, as predicted by the particular mathematical form of the orbital.

With this in place:

*

*Levine and Wikipedia part 1 are in complete agreement. Wikipedia is a more detailed (but less precise and more talkative) description of the same concept.

*Wikipedia part 2 presents notation which (i) is indeed used in introductory textbooks, but which (ii) is not used in any professional capacity in research or engineering in quantum mechanics.

What orbitals really are is wavefunctions $-$ this is what the term is understood to mean in the full theory of quantum mechanics. And, as wavefunctions, orbitals are also associated with probability distributions (though it's important to remember that the wavefunction carries more information than just the probability distribution), and those probability distributions are similarly associated with the spatial regions where they are supported.
In introductory texts it is sometimes useful, for didactic purposes, to identify the orbital with this spatial region, and you can sometimes get relatively far on this notion, but it is important to keep in mind that this is a 'lie to children' and that in the full theory 'orbital' implies a wavefunction.
A: (Disclaimer: I am only a highschool student and have learned the following mostly on my own. If there are any mistakes, please feel free to correct me!)

An atomic orbital represents the probability distribution* of the location of an electron around the nucleus and is mathematically described by a wave function.
Now what does this mean? Let's start with what an atomic orbital isn't:

*

*An orbital is not a fixed spatial region or a "container" in which an electron can move around - In Quantum mechanics, an electron does not have a specific location.

So what is an atomic orbital?

*

*As mentioned before, the electrons don't have a fixed position (and momentum, but this seems less relevant to me at this point), so we cannot determine its position to a single point - this only happens when we measure the position.


*When we measure the position, we find it to be more likely to be present at some points than at other points. This is what is meant by the probability distribution - it simply describes the probability of "finding" an electron when measuring its position for every point in space. So theoretically, there is a probability that at any point in time, some electron is 100km away from the atom it belongs to, but this probability is extremely small. (see What is the probability for an electron of an atom on Earth to lie outside the galaxy?)


*Now assume that we measure the position of the electrons for 1000 times and plot the measured positions to some 3-dimensional model of our atom. We will find that in 90% of the cases the electron is in a certain area of space and this is usually depicted by the familiar atomic orbital shapes:

(Source)
So the shapes of the orbitals as they are most often depicted is usually chosen in such a way that the probability of finding the electron inside this shape (when measuring its position) is at least 90%. However, note that the electron is not constrained to this shape and there is a probability that it is measured outside.
There are some other things to mention about orbitals apart from their "shape". One of these is that every orbital has a certain energy level associated with it. This means that when an electron is in an orbital $A$ it has the exact energy associated with $A$.
If there is another orbital $B$ with higher energy level than $A$, the electron in $A$ can "jump" to $B$ if it absorbs the exact amount energy which is the difference between the energy levels of $A$ and $B$. The most common example is an electron absorbing a photon which has the wavelength that corresponds to the energy differents of the orbitals. Likewise, electrons can jump to an orbital with lower energy by emitting a photon with the wavelength corresponding to the difference in energy between the orbitals.
Here is a graph showing the relative energy levels of some atomic orbitals:

(Source)
I hope this somewhat clears up the confusion.

*As mentioned in the comments, the wavefunction $\psi$ describing an atomic orbital does not directly give the probability density, but the probability amplitude. The probability density can be obtained by $|\psi |^2$ for complex orbitals or $\psi ^2$ for real orbitals.
A: If you take any linear solution $\Psi(r,\theta,\phi)$ to Schrödinger's Equation in 3 dimensions (spherical coordinates $(r,\theta,\varphi)$) and a probability $P = \vert \Psi \vert^2$, representing the wave function of your atomic orbital, you can "split it" in both radial and angular functions:
$$\Psi(r,\theta,\varphi) = R(r)Y(\theta,\varphi)$$
(note that $R$ and $Y$ implicitly depend on atomic numbers, so are different for different atomic orbitals).
Then the representation we have of atomic orbitals is a 3-D plot of both radial probability density $$D_r = r^2\cdot R^2(r)=\frac{\mathrm{d}P(r)}{\mathrm{d}r}$$ and angular probability density $$D_a = Y^2(\theta,\phi) = \frac{\mathrm{d}^2P(\theta,\varphi)}{\sin\theta \mathrm{d}\theta\mathrm{d}\varphi}$$
evaluated and plotted in spherical coordinates around your atom.
A: It's important to note the atomic orbitals are approximations. In the context of the basic hydrogen atom Schrödinger Equation, they are exact eigenstates of energy, total angular momentum squared, and $L_z$, where $z$ points in any direction you want it to.
As energy eigenstates, they are stationary states, and their time evolution involves a global phase rotating with frequency $E/\hbar$. As such, they can never change, which obviously contradicts experiment. Call this "problem 1".
Also: in quantum mechanics, the electron is a point particle. This leads to problematic interpretations that have their uses, but are not fundamental. One of these interpretations is that electron moves randomly in a fashion that has it inside an orbital boundary 90% of the time. Call this "problem 2".
Both these problems are addressed in quantum field theory, in which the electron is no longer a point particle, but the minimum excitation of the electron field, a spinor field that fills all space. With that, an orbital describes how the electron field excitation of a single electron is spread out over space in an approximate energy eigenstate, and how it propagates in time.
The wave function then represents the complex quantum amplitude, whose modulus squared is the probability density of the electron's location. There really is no intuitive (or classical) way to understand coherent complex amplitudes of fermion fields, other than it's kind of like how we treat light...but with conserved quantum numbers, antiparticles, and Fermi-Dirac statistics.
The quantum field treatment also applies to the electromagnetic field, which then adds an interaction term to the hamiltonian, and allows transitions between states. It also adds virtual electron positron pairs to the binding, and that's only at the 1st order. The actual complexity of the state is beyond calculation.
With that, I would say the wave function is a mathematical approximation to something physical. I do believe this conundrum is the origins of Feynman's two famous quotes on quantum mechanics:
The disheartening,
" I think I can safely say that nobody understands quantum mechanics."
and the practical,
"Shut-up and calculate"
