Representation theory studies (among other things) representations of groups by finite matrices. The non-commutative analog of classical Fourier transforms.
Representation theory is a tremendously important area of pure mathematics. By representing elements of algebraic structures as linear transformations, we can study that algebraic structure as well as modules over these structures.
It reduces problems in abstract algebra to tools in linear algebra, which is a better understood.
Representation theory generalizes Fourier analysis to harmonic analysis, and is also used in the study of automorphic forms in number theory.