Assume we want to define $\mathrm{Tor}_n (M,N)$ where $M,N$ are $R$-modules and $R$ is a commutative unital ring.
We take a projective resolution of $M$:
$$ \dots \to P_1 \to P_0 \to M \to 0$$
Now does it matter whether we apply $-\otimes N$ or $N \otimes -$ to this? It shouldn't because we have $N \otimes P \cong P \otimes N$. Right? Thanks.
I guess the point here is that $N \otimes_R -$ and $- \otimes_R N$ are naturally isomorphic functors. Therefore, you get an isomorphism of chain complexes $N \otimes_R P^{\bullet} \cong P^{\bullet} \otimes_R N$, which implies that the two complexes have isomorphic homology. So, it doesn't matter if you apply $N \otimes_R -$ or $- \otimes_R N$.