The Hohenberg-Mermin-Wagner theorem predicts that, due to fluctuations in low-dimensional systems, no local order parameter may be used to signal a phase transition at finite temperatures and that no true long-range order takes place. However, in two dimensions, a Berezinskii-Kosterlitz-Thouless phase transition takes place, where two different regimes may be characterized by different space dependences of some correlation functions. The mechanism in a spin system is identified and is the consequence of spin vortices whose creation and proliferation at high temperatures lead to regimes of quasi-long-range order with a power law decay or an exponential decay, depending on the phase. In addition to some simplified discussion of the transition, a more detailed study is presented using the renormalization group approach.