Surjective morphism to structure sheaf $\mathcal{O}_X$

Question

Let $X$ be a non-singular algebraic variety, $\mathcal{O}_X$ the structure sheaf. We know that an inclusion of $\mathcal{O}_X$-modules

$0\rightarrow \mathcal{F}\rightarrow \mathcal{O}_X$

implies that $\mathcal{F}$ is an ideal sheaf. I want to know if we have a surjective morphism

$\mathcal{F}\rightarrow \mathcal{O}_X\rightarrow 0$

then what information of $\mathcal{F}$ can we get?