In the categories of topological spaces, rings, groups and sets I know that a morphism is a monomorphism iff it's injective.
Things are different for schemes. In fact I know that a scheme injective morphism isn't in general a monomorphism (and I have some counter examples), but I don't know whether the converse is true of false.
A monomorphism between schemes is injective?
On
By default, an injective morphism of schemes is an injective map of the underlying topological spaces.
A morphism of schemes $f:X\to Y$ is a monomorphism if and only if the diagonal morphism $\Delta_{X/Y}:X\to X\times_Y X$ is an isomorphism, see Tag 01L3.
A morphism of schemes $f:X\to Y$ is universally injective if and only if the diagonal morphism $\Delta_{X/Y}:X\to X\times_Y X$ is surjective, see Tag 01S4.
So we have
$\ \ \ \ $[$f$ is a monomorphism]
$\Leftrightarrow$[$\Delta_f$ is isomorphic]
$\Rightarrow$[$\Delta_f$ is surjective]
$\Leftrightarrow$[$f$ is universally injective]
$\Rightarrow$[$f$ is injective]
So the answer is yes, any monomorphism of schemes is injective.
a quick google search for "monomorphisms of schemes" yields this nice post:
https://mathoverflow.net/questions/56591/what-are-the-monomorphisms-in-the-category-of-schemes
Condition (c) at least answers your question in the positive for morphisms locally of finite type.
For arbitrary affine schemes note that a monomorphism of Spec's corresponds to an epimorphism (ie, surjection) of rings. For a surjection of rings, it's clear that the induced maps on spec (by taking preimages of primes) must be injective. Here you don't need the LoFT condition.
Passing to arbitrary schemes and arbitrary monomorphisms would probably require a more subtle analysis. A general technique in this area is essentially using the idea that any morphism of schemes is an inverse limit of LoFT morphisms (this might not quite be correct - I don't have a precise statement).