[1][2] Indecomposable is a weaker notion than simple module (which is also sometimes called irreducible module): simple means "no proper submodule" N < M, while indecomposable "not expressible as N ⊕ P = M".
This is the case for modules over a field or PID, and underlies Jordan normal form of operators.
Explicitly, the modules of the form R/pn for prime ideals p (including p = 0, which yields R) are indecomposable.
A module of finite length is indecomposable if and only if its endomorphism ring is local.
Still more information about endomorphisms of finite-length indecomposables is provided by the Fitting lemma.