In interpretations of QM where the wave function is real, what does that mean? In a lot of interpretations of Quantum Mechanics they believe that the wave function is "real". But what does that mean? Are they saying that the wave function of an elementary particle (electron/photon) is a real wave, like a water wave, oscillating in space time?
 A: No one thinks the wavefunction of nonrelativistic quantum mechanics is a wave in space and time. No one that wants to agree with observations that is.
If someone is a realist about the wavefunction of nonrelativistic quantum mechanics then they start out by making a mathematical model by having the model include a wavefunction. Which when there are $n$ particles is a function from $\mathbb R^{3n}$ into a tensor product of the spin states of every particle in the universe (there are other alternatives that are mathematically just as good). They would identify two waves that are the same except an overall nonzero complex scalar multiple.
They would then postulate that it evolves according to some dynamical laws, such as the Schrödinger equation.
They would then explain how features of the mathematical model correspond to experimental set ups and the results of experiments and observations.
Just like Newtonian mechanics might model things as a path in configuration space that satisfies a dynamical law (and maybe some other principles for when the favored dynamical laws fail) and then explain how features of the mathematical model correspond to experimental set ups and the results of experiments and observations.
