I'm trying to a build a better understanding of presentations. I get that a a group has a presentation $\langle S \mid R \rangle$ if it is the "freest" group subject to the relations $R$. 

But, for instance, how is it immediately clear that $\langle x, y \mid x^2 = y^2 \rangle$ is the same group as $\langle x, y | xyx^{-1}y \rangle$? Also, more generally, what are some ways to determine if two different presentations yield the same group?