To demonstrate the power of character theory, we give a couple of examples in group theory where the statement of the theorem does not involve characters, but where they are used in the proof.
Character theory was developed before the foundations in terms of representations were made clear by Frobenius' student Schur. Frobenius invented characters of nonabelian groups (after Dirichlet had invented characters of abelian groups and applied them in number theory) before 1900. Burnside in England was a professor of engineering mathematics at the naval academy, and thus an applied mathematician. Group theory had nothing to do with his `real' work of educating shipbuilders, and Frobenius probably regarded him as an amateur, but Burnside's discoveries were also fundamental. After Frobenius, Burnside and Schur, one mathematician should be mentioned as developing character theory to its current state of perfection, which is Richard Brauer. Brauer belonged to a different generation – he began working in Germany in the 1930's with Hasse and E. Noether, and came to Princeton in 1934 to escape the Nazis. His most important invention is modular representation theory, in which the field has characteristic $p$.