This is a special case of the general Krull-Schmidt-Remak-Azumaya theorem.
Theorem 2.12 (Krull-Schmidt-Remak-Azumaya Theorem) Let $M$ be a module that is a direct sum of modules with local endomorphism rings. Then any two direct sum decompositions of $M$ into indecomposable direct summands are isomorphic.
There is an elementary proof of the special case based on a generalization of the concept of dimension of vector spaces, which you can find, for instance, in Jacobson's “Basic Algebra II”.
The condition is equivalent to the property that there is some cardinal $\kappa$ such that every left $R$-module is a direct sum of modules with at most $\kappa$ generators.
Theorem 2 of
Warfield, R. B. jun, Rings whose modules have nice decompositions, Math. Z. 125, 187-192 (1972). ZBL0218.13012.
shows that the commutative rings with this property are precisely the Artinian principal ideal rings.
For not necessarily commutative rings, Theorem 2.2 of
Griffith, P., On the decomposition of modules and generalized left uniserial rings, Math. Ann. 184, 300-308 (1970). ZBL0175.31703.
shows that all rings with this property are left Artinian.
There is a property of rings called (left) pure semisimplicity, which has several equivalent definitions. Section 4.5 of
Prest, Mike, Purity, spectra and localisation., Encyclopedia of Mathematics and its Applications 121. Cambridge: Cambridge University Press (ISBN 978-0-521-87308-6/hbk). xxviii, 769 p. (2009). ZBL1205.16002.
gives a good survey, including such results as:
(Theorem 4.5.4) A left pure semisimple ring is left Artinian, and every left $R$-module is a direct sum of indecomposable finite length indecomposable modules.
(Theorem 4.5.7) $R$ is left pure semisimple if and only if every left $R$-module is a direct sum of indecomposable modules, if and only if there is a cardinal $\kappa$ such that every left $R$-module is a direct sum of modules of cardinality less than $\kappa$.
(Note that one consequence of all of this is that if every left $R$-module is a direct sum of modules with at most $\kappa$ generators for some $\kappa$, then in fact every left $R$-module is a direct sum of finitely generated modules.)
So a complete answer to the question is that the rings in question are precisely the left pure semisimple rings, although one might argue that this is just replacing the original condition with an equally mysterious one.
But there is a conjecture (the Pure Semisimplicity Conjecture, which is 4.5.26 in Prest's book) that the left pure semisimple rings are precisely the rings of finite representation type (i.e., for which every module is a direct sum of indecomposable modules, and with only finitely many indecomposables). It is known that every ring of finite representation type is left pure semisimple. Also, "finite representation type" is a left/right symmetric property, so if the conjecture is true then pure semisimplicity is a left/right symmetric property.
Best Answer
Well in basic Wedderburn theory, given an $R$ module $M$, you call the sum of all simple submodules of $M$ which are isomorphic to a simple $R$ module $S$ a "homogeneous component of $M$".
For semisimple (Artinian) rings, the homogeneous components of the ring itself are exactly the simple rings in the Wedderburn decomposition.
I suppose one property you could draw from this is that their endomorphism rings are full linear rings over a division ring. (The division ring being the division ring of endomorphisms of $S$, which is common to all of the modules.)
And as others have noted they are obviously semisimple. What is nicer than semisimple? (Besides simple? and f.g. semisimple?)