Let $X$ be a scheme. I was trying to think of an intrinsic (or coordinate free, if you like) way to characterize all affine open $U\subset X$. For instance, something like "$U$ is open and points of $U$ correspond bijectively to prime ideals in $O_X(U)$ via $$p\mapsto \{f \in O_X(U): \text{the germ of $f$ at $p$ is contained in the maximal ideal of}~ O_{X,p}\}.$$
I don't know if that works, but is there some such characterization?