For instance say that $X$ is a degree-$d$ hypersurface in $\mathbb{P}^n$. To be precise, put $R = k[x_0, \ldots, x_n]$, $d > 0$, $f \in R_d$ squarefree, and $X = V(f) \subset \mathbb{P}^n$. Then we have an exact sequence of graded $R$-modules
$0 \to R(-d) \xrightarrow{-\cdot f} R \to R/(f) \to 0$;
when we promote this to sheaves, apparently we have
$0 \to \mathcal{O}_{\mathbb{P}^n}(-d) \to \mathcal{O}_{\mathbb{P}^n} \to \mathcal{O}_X \to 0$.
Now normally I'd think of "$\mathcal{O}_X$" as representing a sheaf on $X$, but this construction seems to have produced a sheaf on $\mathbb{P}^n$ instead.