Given two sets $A$ and $B$ such that $A \subset B$ What morphism do I use to express this relation in a category of sets.
Is it $ i : A \to B$ where $i$ is the inclusion map? or $ pr : B \to A$ where $pr$ is the projection map that "forgets" $B-A$?
Also what is the relation of those two maps? Are they inverse to each other? If not what is the inverse map of $i$?
The generalisation of "subset" in a category is given by the concept of a subobject, which is a collection of monomorphisms $A \rightarrowtail B$ intuitively sharing the same image in $B$. In the category of sets, a subobject corresponds to a subset up to isomorphism (if $A$ is an actual subset of $B$, and $A'$ is isomorphic to $A$, we still want to count $A'$ as a subset of $B$). Every inclusion $i : A \hookrightarrow B$ identifies a subobject.
For the second part of your question, projection maps are not unique: perhaps you could clarify what map $B \to A$ you are considering? An inclusion $i : A \hookrightarrow B$ does not generally have an inverse (though there is a natural inverse image function $i^{-1} : B \to \mathcal P(A)$).