= Solution
A <group representation> on a <vector space> $V$ is a <group homomorphism> $R:G\to\operatorname{GL}(V)$: $R(gh)=R(g)R(h)$ and $R(e)=I$. A <reducible representation> has a nonzero proper <invariant subspace> $W\subset V$, meaning $R(g)W\subseteq W$ for every $g$. Over the <complex numbers>, a finite-dimensional <representation> of a <finite group> is a <direct sum> of <irreducible representations>, by <Maschke's theorem>.
The <character of a representation> is $\chi_R(g)=\operatorname{tr}R(g)$. The cyclic property of the <trace> makes it a <class function>:
$$
\chi_R(hgh^{-1})=\operatorname{tr}\bigl(R(h)R(g)R(h)^{-1}\bigr)=\chi_R(g).
$$
Thus a <character> is specified by one number per <conjugacy class>, much less information than all the <representation> <matrices>. Moreover,
$$
\chi_{R\oplus S}=\chi_R+\chi_S,\qquad
\chi_{R\otimes S}=\chi_R\chi_S,\qquad
\boxed{m_r=\frac1{|G|}\sum_{g\in G}\overline{\chi_r(g)}\chi_R(g).}
$$
Here <character orthogonality> says that $m_r$ is the multiplicity of the <irreducible representation> $r$. Consequently <characters> determine finite-dimensional complex <representations> up to <equivalence of group representations> and compute <tensor product of group representations> decompositions without constructing a new <basis> explicitly.
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