The following structure theorem is well known:
A commutative Artinian ring is a finite direct product of local Artinian rings.
Do we have such/similar structure theorem for noncommutative artinian rings?
I think the best analogy you can get is:
Basically, one arrives at the result you mentioned in the OP by noting that you cannot have an infinite chain of pairwise orthogonal idempotents in a commutative Artinian ring.
Every finite set of central orthogonal idempotents of $R$ which add up to 1 corresponds to a decomposition of the ring into factor rings $e_iRe_i$.
If a central idempotent $e$ cannot be written as the sum of two central orthogonal idempotents, then $eRe$ is indecomposable (in the sense of ring decompositions). Let's call this idempotent $e$ irreducible, for the duration of our conversation.
In any ring, you can try to write $1=\sum e_i$ as a sum of central irreducible idempotents, but sometimes irreducible ones do not exist, and sometimes these sets can be infinite. Under pretty mild finiteness conditions you can guarantee the existence of a finite set of central irreducible idempotents. Even "Noetherian" works, of course.
So why does "local" show up in the commutative case? Lemma: a commutative Artinian ring whose only idempotents are 0 and 1 is local. (Hence a commutative Artinian indecomposable ring is local.)