The title tells everything: an abelian group object in a category $\mathbf C$ with finite products is a triple $(G,m,e)$, $m\colon G\times G\to G$, $e\colon 1\to G$ such that the well known diagrams commute and such that $m\circ \sigma_{GG}=m$, where $\sigma_{AB}\colon A\times B\to B\times A$ is the map turning $\mathbf C$ into a symmetric monoidal cat.
Thanks in advance!
On
A topos is a category of sheaves on some site. The abelian group objects are then the category of sheaves of abelian groups on this site. The category of sheaves of abelian groups on a site is an abelian category.
[I wouldn't use the phrasing "abelian subcategory of $\mathcal E$" since $\mathcal E$ itself has no abelian category structure, and the abelian group objects are not a full subcategory (not all morphisms of sheaves will respect the abelian group structure on the sheaves of abelian groups).]
I will only assume that $\mathcal{E}$ is an elementary topos; this includes the case of toposes of sheaves over a site. Let $\textrm{Ab}(\mathcal{E})$ be the category of abelian group objects in $\mathcal{E}$. It is indeed true that $\textrm{Ab}(\mathcal{E})$ is an abelian category: to show this, one merely repeats in the internal logic of $\mathcal{E}$ the usual argument that $\textrm{Ab}(\mathbf{Set})$ is an abelian group. The only thing to beware of is that the internal logic of $\mathcal{E}$ is intuitionistic, i.e. the law of excluded middle does not necessarily hold in the internal logic $\mathcal{E}$. But this is no great obstacle as the usual arguments are constructive (as one would hope for such elementary facts)!
For the case where $\mathcal{E}$ is a more general category, I refer you to this MathOverflow question.