irreducible representation
Irreducible representations are the atoms of representation theory. Just as every integer factors into primes, in good situations every representation breaks into irreducible pieces; understanding the atoms means understanding everything built from them. An irreducible representation is one that cannot be cut down any further while respecting the group action.
Formally, a representation V of G is irreducible (or simple) if V is nonzero and its only G-invariant subspaces are 0 and V itself. A G-invariant subspace W is one with rho(g)W contained in W for all g — that is, W is preserved by every group element. Equivalently, V is irreducible exactly when it is a simple module over the group algebra k[G]: it has no proper nonzero submodule.
Whether a representation is irreducible depends sharply on the field. The standard 2-dimensional real representation of the rotation group of a regular n-gon is irreducible over R (no real invariant line), yet becomes reducible over C, splitting into two complex lines. So irreducibility is a statement about V together with k, never about the abstract group alone.
S_3 has exactly three irreducible complex representations: the trivial one (degree 1), the sign representation (degree 1), and the standard representation (degree 2). The dimension sum of squares is 1 + 1 + 4 = 6 = |S_3|.
The sum-of-squares rule pins down the degrees of all irreps of S_3.
Over an algebraically closed field of characteristic 0, the number of irreducible representations of a finite group (up to isomorphism) equals the number of conjugacy classes — a clean bookkeeping fact with no obvious natural bijection.