Consider some space $X$. Does the fundamental group tell us information about the equivalence between two paths $f,g: I \to X$? So there exists a homotopy $h: I \times I \to X$ such that $h(s,0) = f(s)$, $h(s,1) = g(s)$, $h(0,t) = x$ and $h(1,t) = y$?
In other words, starting with two paths, how/why do we introduce the fundamental group?