One thing I’ve noted about this thread is nobody has referred to the probability model distribution of electrons as “electron clouds”. That’s how I was taught to understand it, and it makes a lot of sense to me.
‘Electron clouds’ is nice because it suggests the fuzzyness of probability, but unfortunately it also implies many electrons (a cloud of electrons) when of course an orbital can consist of a single electron.
Q.E.D. great posts, thanks.
As for understanding photons, it’s a bit like understanding anything, the more you really look into the subject, the more you will find that you don’t understand. If you accept that you cannot think of them in terms that would be sensible for normal sized objects then you will find it easier.
For instance the quantum world you cannot assume that if something can be detected at place a, then later at place b, the thing necessarily ever existed in any place between a and b.
In case anyone is interested, here is a very basic description of how said clouds are generally drawn, based on their probability wave functions. At least, this is how much I understand it, but it should give you a grasp of the very basics, which is what I feel I have.
In the case of the first 2 electrons in an atom (only one in hydrogen, 2 in helium, and the first 2 in any of the other atoms), the “cloud” is generally understood to be a sphere. It is often called an “S-orbital”. This means that the particular electron you are observing has the highest probability of being anywhere inside that sphere at time X, including, according to the math but contrary to many mental images, inside the nucleus itself. There is a probability that it will be outside the sphere, but it is very small - this is explained by the math behind all of this.
the 3rd and 4th electrons (assuming the atom has as many) are likely to be found inside an area that has a dumbell like shape - or perhaps more accurately, an elongated infinity symbol, called the P orbital. These electrons can be found in either of the two lobes, though oddly enough it can never be inbetween, at the intersection of the lobes. Again, this is mathematical, and hard to conceptualise. but there it is anyways! The same electron (say, the 3rd one, found alone in that orbital in a lithium atom) can and will be found in both the lobes, though how it can travel from one to the other is hard to understand (if it has been figured out at all?)
The 5th and 6th electrons share another dumbell-shaped orbital, which is located perpendicular to the other one. The same is true for the 7th and 8th electrons - the three dumbell orbitals are often refered to as the Px, Py and Pz ones since they can easily be drawn relative to one another the way we draw the three dimensions on a set of axes.
After that, it gets a little more complicated - the d and f orbitals have shapes that I can’t describe right now (mainly because I can’t remember them well enough, and because the multitudes of them get really complex). An atom is comprised of as many orbitals as it takes to accomodate all it’s electrons.
We usually think of the electrons as being in “shells” like orbits, which works fairly well to try and understand the whole energy-level concept. For example, lithium has three electons -two in the S sphere, and 1 in a P orbital. This is the most stable array of electrons, since the third one cant get closer “in” since it is blocked by the first two (remember electrons repell each other too!) When it gets energised (say, from a photon hitting it) it might be bumped out of that stable orbital, into a space it doesnt want to be in, “ouside” the dumbell. The energy it took to do that was equivalent to the energy of the photon that hit it - it absorbed the photon. The electron then spontaneously falls back to where it is “comfortable”, but to do that it has to get rid of the energy it now has, and so it “throws it away”, in the form of another photon. Therefore, light is emitted, and the atom is back to how it was, nice and happy.
Now I hope I didnt make a gargantuan mistake in all that - though it’s likely 
“Mathematical description” is basically correct. How you view it may be dependent on your philosophy of Science - whether equations & theories are near as can be describing the actual phenomena - (‘realism’), or these are simply tools that are useful models - (‘instrumentalism’), or some other view.
I would say that the use of sine & cosine functions (which I assume to be the meaning of ‘wave’ there) is to a small extent a matter of convenience, but if you just meant ‘look like’ that’s basically correct.
In fact, there’s a current thread Electron Orbital Radius in which you can see the wave function for a very simple case (an electron in a hydrogen atom). In other cases the mathematics gets extremely complicated although the general idea can be seen (in the same way that describing the exact movements of a swimming school of fish is highly complicated but you can see what’s going on).
(Just stop worrying, and learn to love the universe.)