Quotient Objects in $\mathsf{Grp}$ I don't know how to precisely formulate my question, but here goes:
Subobjects and quotient objects are duals, so a quotient object in $\mathsf{Grp}^{\text{op}}$ is a subobject in $\mathsf{Grp}$. The arrows of these categories are in bijection, and the subobjects of a group $G$ in $\mathsf{Grp}$ are in bijection with the subgroups of $G$. I'm guessing that generally speaking, the subobjects of $G$ are not in bijection with its quotient objects.
Nowhere do normal subgroups arise, so I'm trying to understand where they fit in this scheme of subobject and quotient object in $\mathsf{Grp}$. I'd expect there to be "more" subobjects (monics) than quotient objects (epics) in $\mathsf{Grp}$ since being a normal subgroup is more restrictive than just being a subgroup.. I'm just overall confused.
 A: What you're saying is right. The subobjects are not in [a reasonable, natural] bijection with the quotient objects; one way you could associate a quotient object $G \to Q$ to each subobject $H \to G$ is to take the quotient by the smallest normal subgroup generated by $H$, but this is not one-to-one.
A: One has to distinguish the property of being a monomorphism from the property of being the kernel of another morphism. These two are the same in any abelian category (by definition, and this allows for the usual correspondence between subobjects and quotient objects by taking kernels and cokernels), but in general, though any kernel morphism is a monomorphism, the converse does not hold.
For example, in the category $\textsf{Grp}$ of groups, the monomorphisms are the injective group homomorphisms, but a monomorphism arises as the kernel of another morphisms precisely if its image is a normal subgroup. For epimorphisms, you don't have this distinction in $\textsf{Grp}$, as any epimorphism (=surjective group homomorphism) is the cokernel of its kernel.
A: The part about the opposite category is probably not so useful for thinking about these things, simply because that category is not very similar to the original category.
You ask where the normal subgroups arise when speaking about subobjects. Well, they are precisely those subobjects of some group $G$, which are kernels (remember that a morphism $f: H\to G$ is a kernel for a map $g: G\to K$ when $gf = 0$ and $f$ is terminal among such morphisms, so whenever we have another morphism $h: H'\to G$ with $gh = 0$ there is a unique morphism $k: H'\to H$ with $h = fk$).
