subrepresentation
A subrepresentation is a smaller representation living inside a bigger one — a slice of the vector space that the group never pushes outside of. If you think of a representation as a room of vectors being shuffled by the group, a subrepresentation is a wall-to-wall sub-room that stays self-contained under every shuffle.
Concretely, let (V, rho) be a representation of G. A subspace W of V is a subrepresentation if it is G-invariant: rho(g)W is contained in W for every g in G. Then restricting each rho(g) to W gives a well-defined representation (W, rho restricted), the subrepresentation on W. Dually, the quotient space V/W inherits a representation, the quotient representation, since the action descends through the projection.
Subrepresentations are exactly the submodules of V viewed as a k[G]-module, so the lattice of subrepresentations is a lattice of submodules. A representation with no subrepresentations other than 0 and V is precisely an irreducible representation; the existence of a proper nonzero subrepresentation is what reducibility means.
Let S_3 permute coordinates of k^3 by rho(g)(x_1, x_2, x_3) = (x_g(1), ...). The line spanned by (1, 1, 1) is invariant (it is the trivial subrepresentation), and the plane x_1 + x_2 + x_3 = 0 is invariant too (the standard subrepresentation).
The permutation representation of S_3 splits into a line plus a plane.