Representation theory studies (among else) 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 and linear algebra is a very developed area.
Representation theory generalises Fourier analysis to harmonic analysis, and representation theory is used in automorphic forms in number theory