Let $C_{\bullet}, D_{\bullet}$ be two nonnegatively graded chain complexs of $R$-modules with maps $d^C,d^D$ respectively($d^C_n: C_{n+1} \to C_n$), and let $f,g: C_{\bullet} \to D_{\bullet}$ be two chain maps. Then, a chain homotopy from $f$ to $g$ is a sequence of $R$-linear maps $h_n: C_{n-1} \to D_{n}$ such that $$f_n - g_n = d^D_{n}h_{n+1} + h_{n}d^{C}_{n-1}.$$
First of all, is my definition correct? My second question is, what does this really mean? My notes say that if such a chain homotopy exist then we say that $f$ and $g$ are chain homotopic and we write $f \cong g$. I feel that the definition is some kind of "isomorphism" between two chain maps but I cant see how, can someone explain what it really means to have a chain homotopy, i cant really "unfold" the definition.