Suppose we have a category (that is, a collection of objects and arrows ["morphisms"] between those objects; for example, the collection of abelian groups and all homomorphisms between those abelian groups). An object $I$ is injective if whenever we have an injection $i:A \to B$ and a homomorphism $f:A \to I$, then there is a morphism $g:B \to I$ such that $f = g \circ i$.
In the category of abelian groups, the divisible groups are precisely the injective objects in that category. To see a sketch of why, suppose $B$ is an infinite cyclic groups, and $A$ is the set of powers of $n$ resting within that cyclic group (hence also isomorphic to an infinite cyclic groups). Suppose $B$ has generator $b$, and $A$ has generated $a:=b^n$. Then a homomorphism $\varphi:A \to I$ amounts to choosing an element of $I$ where $a$ will be sent. Extending this to a homomorphism from $B$ to $I$ amounts to choosing an element of $I$ whose $n$th power is $\varphi(a)=\varphi(b)^n$; that is, dividing by $n$. This is why they are related.
Now, you might ask, why are injective objects important? Injective objects are some of the most important objects in a category which underlie many applications of homological algebra. Suppose we have an exact sequence $0 \to A \to B \to C \to 0$. It is well known that the sequence $0 \to Hom(C,I) \to Hom(B,I) \to Hom(A,I)$ is exact no matter what. However, we know that that last map is surjective if $I$ is injective. Thus, $Hom(-,I)$ is an exact (contravariant) functor iff $I$ is injective. It follows from this that $Ext(-,I)$ is trivial iff $I$ is injective, and in general, right derived functors are trivial on injective objects.
This latter fact is interesting because derived functors appear all over higher mathematics. To give an example, the group cohomology is always trivial on injective objects in the category of $G$-modules. However, all we talked about was injective objects in the category of $\mathbb{Z}$-modules (i.e. abelian groups). So why is this interesting? Well, the functor which sends an abelian group $M$ to the induced $G$-module $\mathrm{Ind}^G(M)$ (essentially $\mathbb{Z}[G] \otimes_{\mathbb{Z}} M$ by extension of scalars) is exact and sends injectives to injectives. This means that we get a whole host of injective $G$-modules by considering injective abelian groups, and what's more, it allows us to show that the category of $G$-modules has enough injectives (i.e. that every $G$-module can be embedded in an injective $G$-module).