We investigate the property of commutator-simplicity in algebras from both algebraic and analytic perspectives. We demonstrate that a large class of algebras possess this property. As an analytic analog, we introduce the concept of topological commutator-simplicity for Banach algebras and establish that a σC∗L1(G,ω)ω-unital σC∗L1(G,ω)ω-algebra is topological commutator-simple if and only if its multiplier algebra is. Furthermore, we explore the applications of commutator-simplicity to certain equations involving commutators, emphasizing its relevance in the study of derivations. Specifically, we obtain that every continuous local derivation on σC∗L1(G,ω)ω is a derivation when G is a unimodular locally compact group with a diagonal bounded weight σC∗L1(G,ω)ω.