character
A character is a clever compression of a whole representation into a single function on the group, obtained by taking the trace of each matrix. It feels like throwing away almost everything — a matrix shrinks to one number — yet, miraculously, over the complex numbers a character remembers the entire representation up to isomorphism. It is the representation's fingerprint.
Given a finite-dimensional representation (V, rho) of G, its character is the function chi: G -> k defined by chi(g) = trace(rho(g)). Two facts are immediate: chi(e) equals the dimension of V, and because trace is invariant under conjugation, chi is a class function — chi(hgh^{-1}) = chi(g), so it depends only on conjugacy classes. Equivalent representations have equal characters, since trace is a similarity invariant.
Over C the character is astonishingly powerful: two complex representations are isomorphic if and only if they have the same character; a representation is irreducible if and only if the inner product of its character with itself equals 1; and the irreducible characters form an orthonormal basis of the space of class functions. Characters thus turn hard questions about representations into linear algebra over a finite-dimensional inner product space.
For the 2-dimensional standard representation of S_3, the character takes value 2 on the identity, value 0 on each transposition, and value -1 on each 3-cycle. Computing the inner product gives (1/6)(1*4 + 3*0 + 2*1) = 1, confirming irreducibility.
A character is determined by its values on conjugacy class representatives.