Giordon answers the question with a wonderful explanation. A related question may also be of interest. The hydrogen atom is an excellent example of a system that can easily be understood as an electron orbiting around a proton. However, there are many caveats.
Here, I’m not supporting the Bohr theory. Instead, there is a probability current density that flows around the proton in circular orbits. Remember that quantum mechanics doesn’t allow you to know where a particle is located (this would require complete ignorance of the particle’s momentum due to the uncertainty principle). We can only talk about the likelihood of finding an electron somewhere.
You may have heard of the charge density and (charge current density) concepts in electromagnetism. This is essentially the idea of the charge density per unit volume and charge flow through a cross section.
These formulas can be understood if we define certain terms. The charge density is what we use to measure the volume. The volume, which is what we use to get to zero. The enclosed charge, which obviously varies depending on the volume. The (charge) current. The area through which current flows. Again, we reduce it to zero. is a unit vector that runs in the opposite direction to the area that defines the current flow. 
Why do I tell you this? It turns out that there are analogous quantities within quantum mechanics. These can be obtained by simply dividing these definitions by the charge. These quantities are known as the probability density or the probability current density in quantum mechanicals. While their meaning is similar to the charge quantities in electromagnetism’s, quantum mechanics calculates these quantities differently. If you don’t have any background in quantum mechanics, the probability density can be described as the absolute square of a wavefunction. The probability current density can be calculated by taking the imaginary portion of the product and derivative of the complex conjugate.
However, you can calculate the probability current density for the electron once you have calculated the wavefunction for an excited hydrogen atom state (the ground state is sphericallysymmetric and does not have a magnetic moment…that would cause the sphericalsymmetry to be broken by choosing a direction to place the magnetic moment along). It is amazing (or unsurprisingly, depending on your perspective), that the probability current flows around a proton in a circular orbit, which matches Bohr’s intuitive model of the hydrogen-atom as a mini-earth/sun system. You can even calculate the magnetic moment that results from this:
- Multiply the charge of an electron by multiplying probability current density to charge current density.
- Integrate to keep the charge current
- Multiply the area of the circular orbit by, to calculate the magnetic moment
It’s clear that images like the one shown below are not as farfetched as many physicists think. However, they’re still quite possible if we’re being honest.
There is a wonderful story that goes along with this regarding how Schödinger originally came up with his famous equation (while on a hike!). If I recall correctly, he was trying to solve exactly this problem (that of the hydrogen atom) and the wavefunction he was using was not the same one we learn in first year quantum classes today. This is because it has an extra factor of the square root of the elementary charge attached to it, as well as a factor of . This is so that the final result for the probability density would have an extra factor of the electron’s charge attached to it—Schrödinger was working with electric charge density, not probability density.
Schrodinger realized the deeper implications of his work only later: charge is distributed probabilistically, but all matter behaves exactly in the same way! 
 Although it is not essential for this answer, I cannot help but mention how the two quantities are related via the continuity equation. This is a statement about the local conservation charge. It is intuitively obvious that the rate at which the charge in an infinitesimal volume changes is equal to that of the charge that flows out of that volume per unit of time.
The time derivative of charge density, on the left, represents the rate at which the charge changes within the infinitesimal volume. On the right, the charge density divided by the time derivative of the current density is minus. This indicates the amount of charge that flows out (hence the plus sign).
You can make the equation look nice by moving the divergence. You can make one side zero and have no minus signs by moving the divergence over to the other.
 The story was told to me by either a Quora user or my first-year quantum professor. I have not seen any official accounts of the event so don’t take this too seriously.
EDIT: Ahhhh…LaTeX disaster. We are working on it.
EDIT 2: It’s all better.