I came across the notion as follows:


Let $X$ be a projective, smooth scheme. And let
$$
 0\to M\to N\to \mathcal{O}\_{X}\to0
$$
be an exact sequence of coherent $\mathcal{O}\_X$-modules. 
What is meant by “the above exact seqence defines an $M$-torsor on $X$”?
I think it may be a standard use of terminology. I just lack knowledge.
Thank you!



notion of torsor defined by exact sequence
From the associated long exact sequence you obtain a map $H^0(X,\mathcal O\_X) \to H^1(X, M)$. $H^1(X,M)$ is just the group paramaterizing $M$-torsors. Since $H^0(X,\mathcal O\_X)$ is free of rank one as a module over itself, this map is equivalent to a single element of $H^1(X,M)$, the image of $1$ - in other words, a single $M$-torsor.


Explicitly one is just viewing the inverse image of $1$ in $N$ as an $M$-torsor.


