Let $R$ be a commutative ring and $M$ and $N$ be $R$-modules (I am not sure if one really needs commutativity in the following). It is well-known that $Ext_{R}^n(M,N)$ for $n>1$ parametrizes $n$-extension of $N$ by $M$, i.e. exact sequences $ 0\rightarrow M\rightarrow C_{1}\rightarrow \dots \rightarrow C_{n}\rightarrow N\rightarrow 0 $ mod certain equivalent relations. Another way to see $Ext_{R}^n(M,N)$ is via derived category; it can be seen as a hom space in $D(R-mod)$ $ Ext_{R}^n(M,N)=Hom_{D(R-mod)}(M,N[n]). $
I now want to understand what $Tor^{R}_n(M,N)$ represent. How should one understand $Tor^{R}_n(M,N)$ intuitively?