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!