This paper introduces the concept of relative homological algebra, focusing on functors analogous to the Tor and Ext functors in Cartan-Eilenberg's work, but adapted to a module theory relative to a given subring of the basic ring. The main goal is to show how certain relative cohomology theories for groups, rings, and Lie algebras can be subsumed under the theory of the relative Ext functor. The paper discusses the definitions and properties of relatively projective and injective modules, and introduces the relative Tor and Ext functors. It also explores the application of these functors to relative cohomology theories for rings and algebras, as well as for groups and Lie algebras. The paper concludes with a discussion of relative cohomology dimension and relative global dimension, and presents a theorem on the cohomology of polynomial rings. The paper provides a detailed analysis of the structure and properties of these functors and their applications in homological algebra.This paper introduces the concept of relative homological algebra, focusing on functors analogous to the Tor and Ext functors in Cartan-Eilenberg's work, but adapted to a module theory relative to a given subring of the basic ring. The main goal is to show how certain relative cohomology theories for groups, rings, and Lie algebras can be subsumed under the theory of the relative Ext functor. The paper discusses the definitions and properties of relatively projective and injective modules, and introduces the relative Tor and Ext functors. It also explores the application of these functors to relative cohomology theories for rings and algebras, as well as for groups and Lie algebras. The paper concludes with a discussion of relative cohomology dimension and relative global dimension, and presents a theorem on the cohomology of polynomial rings. The paper provides a detailed analysis of the structure and properties of these functors and their applications in homological algebra.