So if I get asked what Ext is in the exam and I say it's the functor mapping an $R$-module $M$ to a homology group $Ext^n_R (M, N)$ as follows:
Take a projective resolution of $M$:
$$\dots P_1 \to P_0 \to M \to 0$$
chop off $M$ to get a chain complex
$$\dots P_1 \xrightarrow{d_1} P_0 \xrightarrow{d_0} 0$$ then apply $Hom(-,N)$ to it to get
$$ 0 \xrightarrow{\overline{d_0}} Hom(P_0, N) \xrightarrow{\overline{d_1}} Hom(P_1, N) \dots $$
and define $$ Ext^n_R(M,N) := ker(\overline{d_n}) / im(\overline{d_{n-1}})$$
