For quantum mechanics, a certain property of a subatomic particle, e.g. the spin of an electron, which can be either up or down, is a "superposition of states," and one of the two conditions, e.g. the fact that it has spin up or down, doesn't manifest itself until the situation is experimentally observed. In fact, from what I understand, it does not exist until you observe it.
So what I would like to understand is a very precise thing: does the electron have a real spin or does it only have it when it is observed?
Does the spin of an electron have an a priori value (unknown to me until I observe it), or does it have a superposed state?
As Einstein said to someone, "Do you really believe that the Moon only exists when you look at it?"
I understand that the observation itself changes the state of the electron, but could this mean that its spin goes from up to down as a result of the observation, or vice versa, or does it mean that it just doesn't have a definite spin?
The fact is controversial to me, but if reality were only true when observed there would be paradoxical, or even physically impossible, consequences.
Best Answer
The premise of your question is not correct: we can experimentally observe superpositions, just not directly. There are many examples of this, but maybe the most famous one is the double slit experiment. I encourage you to take a look for yourself, but the key point is that the existence of superpositions manifests itself through the emergence of interference patterns in our experiments. This is why we conjured up the concept of superposition in the first place!
Superpositions are essentially a manifestation of the wave-like bahaviour of quantum particles (again, look at the damn experiment, it's crazy cool!).
Warning, this next part is a little bit more tecnical:
If you really dislike the concept of superposition you could try to just stop imagining particles as discrete objects and start treating them strictly as waves1, but this is hardly productive, it's better to stick to what the experiments tells us: sometimes particles behave like classical dot particles, sometimes they behave like classical waves, and that makes them neither of those two. Quantum particles are, well, quantum particles! Their physical state is well represented by vectors in Hilbert space, so linear objects that support being linearly combined in the basis of eigenstates of observables (remember that QM dynamics is linear), id est support superpositions.
[1]: But there is no escape from wave-particle duality, so prepare to accept that a Dirac's delta is a wave!