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A subrepresentation of ρ: G → GL(V) is a subspace W ⊆ V that is invariant under every ρ(g). A representation is irreducible if its only invariant subspaces are {0} and V itself — it cannot be broken into smaller pieces. Irreducible representations are the atoms of representation theory: under favorable conditions (characteristic zero, finite groups), every representation decomposes as a direct sum of irreducibles.
The idea of breaking a representation into simpler pieces is the heart of the subject. Given a representation ρ: G → GL(V), a subrepresentation (or invariant subspace) is a subspace W ⊆ V such that ρ(g)(W) ⊆ W for every g ∈ G — the action of G keeps W within itself. In matrix terms, if we choose a basis where the first k vectors span W, every ρ(g) takes block upper-triangular form with a k×k block in the top-left corner. That block defines a representation of G on W.
A representation is irreducible (or simple) if its only invariant subspaces are {0} and V. This means there is no way to decompose the action into smaller independent pieces. For a one-dimensional representation, this is automatic. For higher dimensions, irreducibility is a strong condition: it says the group action thoroughly "mixes" the space, so that no proper subspace is left alone by all group elements simultaneously.
When a representation is reducible (has a proper invariant subspace W), the natural question is whether V decomposes as a direct sum V = W ⊕ U where U is also invariant. If so, the representation splits into two independent pieces. A representation is completely reducible (or semisimple) if it decomposes as a direct sum of irreducible subrepresentations. This is not automatic — the example of [[1,1],[0,1]] for ℤ/2ℤ over a field of characteristic 2 shows a reducible representation with no invariant complement.
The remarkable fact, which you will see formalized as Maschke's theorem, is that for finite groups over fields of characteristic zero (or more generally, characteristic not dividing |G|), complete reducibility is guaranteed. This means the study of all representations reduces to the study of irreducible ones plus the combinatorics of how they assemble via direct sums. Irreducible representations are thus the atoms from which the entire representation theory of a group is built — finding and classifying them is the central problem.