I am reading a lot about the theory of special relativity, but I have a very basic question about this theory I still don't understand. Consider a particle in two inertial reference frames $\Sigma$ and $\Sigma'$. The reference frame $\Sigma'$ is moving with uniform velocity $v$ relative to $\Sigma$. The particle is at rest in $\Sigma'$. Both reference frames have common axes $x$ and $x'$. When doing a certain calculation in both reference frames, which one of the obtained results is considered correct? Are they considered both correct or the one obtained by an observer in $\Sigma'$?
They are both correct. You can do physics in any inertial frame you want. (In fact, you can do it in non-inertial frames too, but it’s more complicated.) That’s basically what “relativity” means.
One thing worth mentioning is that not all physical quantities are relative, i.e., dependent on which reference frame you measure them in. There are lots of absolute quantities that are independent of reference frame. These are the Lorentz scalars. For example, the energy and the momentum of a particle are frame-dependent, but the invariant mass is frame-independent.