From http://en.wikipedia.org/wiki/Injective_module

Every submodule of every injective module is injective if and only if the ring is Artinian semisimple (Golan & Head 1991, p. 152); every factor module of every injective module is injective if and only if the ring is hereditary, (Lam 1999, Th. 3.22); every infinite direct sum of injective modules is injective if and only if the ring is Noetherian, (Lam 1999, Th 3.46).

Another characterization of (one sided, as above) noetherianity with injective modules: every injective is direct sum of indecomposables; there is only a set (not proper class) of injective indecomposables. 

Dually one has many characterization results concerning projectivity; for example for a commutative integral domain, one of the many equivalent conditions to be Dedekind (Pr\"ufer) is hereditary (semi-heredirary).

QF rings: every projective is injective (on one side) iff every injective is projective (on one side) iff the ring is two-sided artinian and the finitely generated right and left modules are in natural duality (with the usual dual, hom to the ring of scalars) iff the ring is (one sided) noetherian and in the Galois correspondence "annihilator" between right and left ideals, each element is closed.

Principal ideal artinian rings: every homomorphic image is QF iff finite direct products of matrix rings over CPU rings (rings $R$ with a nilpotent element $p$ such that $pR=Rp$ and the $p^nR=Rp^n$ are all one sided ideals) iff each finitely generated module has a lattice of submodules which is finite direct product of primary lattices. [Generalizing primary decomposable to semiprimary characterizes artinian serial rings]

von Neumann regular: $\forall a\exists x:axa=a$ iff each in each finitely presented (or even only finitely generated projective) module the semilattice of finitely generated submodules is a complemented sublattice of the lattice of all submodules.

Classically semisimple: every module ahs complemented lattice of submodules.

Coherent: Chase theorem

One can multiply examples, ad enjoy finding them in the books by Lam, Rowen, Faith, Stenstr\"om, ...

However: all the magic disappares when one notes that every Morita invariant property can be defined by a purely categorical property of the abelian category of all right modules, and when in such a category one fixes a progenerator then any property of the ring can be expressed in this language (compare semisimple rings, a Morita invariant property, versus skew fields, characterized by freenes of modules instead of injectivity / projectivity).
Besides, all these characterizations are also possible in lattice theoretic terms (Hutchinson - Isbell lattice associated to a abelian category); when the property to be expressed is not Morita - invariant one needs to fix a basis in the lattice.

So the problem is not the possibility to express a ring property in the categorical language of modules (bicartesian language of abelian categories, or the language of lattice theory). The problem is to formalize the requested "elegance" of the characterization, a problem with no clear mathematical answer.