What is Irreducible representation, character table, Mulliken symbol?

A representation that cannot be split further by any change of basis is an irreducible representation. The character table of a point group lists the characters of its irreducible representations, one row each, on its classes, one column each, with the Cartesian functions and rotations that transform like each row. The rows are named by their Mulliken symbol: $A$ or $B$ for one dimension (symmetric or antisymmetric under the principal rotation), $E$ for two, $T$ for three; subscripts 1, 2 (symmetric or antisymmetric under a $C_2$ or $\sigma_v$), $g$, $u$ (under inversion), primes (under $\sigma_h$).