In EGA IV, 17.2.6 the following characterization of monomorphisms is given:
Let $f : X \to Y$ be a morphism locally of finite type. Then the following conditions are equivalent:
a) $f$ is a monomorphism.
b) $f$ is radicial and formally unramified.
c) For every $y \in Y$, the fiber $f^{-1}(y)$ is either empty or isomorphic to $\text{Spec}(k(y))$.
Also note that (due to the adjunction) a morphism between affine schemes is a monomorphism if and only if the associated homomorphism of rings is an epimorphism (in the category of rings) and the latter ones can be characterized in many ways. See, for example,this Bourbaki seminar and this MO discussion.
In EGA IV, 18.12.6 it is shown that proper monomorphisms are exactly the closed immersions. Also note that a morphism $X \to Y$ is a monomorphism if and only if the diagonal $X \to X \times_Y X$ is an isomorphism. In particular, every monomorphism is separated.