Irreducible Representations of Groups
Definition
Let a group $G$ and a finite dimensional vector space $V$ be given. Call $\operatorname{GL}(V)$ the general linear group. The following group homomorphism $\rho$ is called a representation of $G$ on $V$.
$$ \rho : G \to \operatorname{GL}(V) $$
$\rho$-Invariant
A subspace $W$ of $V$ is said to be $\rho$-[invariant] or $G$-invariant if it satisfies the following.
$$ \rho (g) (W) \subset W, \qquad \forall g \in G $$
Subrepresentation
For a $\rho$-invariant $W$, any $\sigma : G \to \operatorname{GL}(W)$ satisfying the following is, plainly, a representation of $G$. Such a $\sigma$ is called a subrepresentation of $\rho$.
$$ \sigma (g) = \rho (g)|_{W}, \qquad \forall g \in G, $$
Irreducible and Reducible Representations
The trivial subspaces $V$ and $\varnothing$ of $V$ are always invariant. If $V$ has a nontrivial invariant subspace, then $V$ is called reducible. Otherwise, i.e., if it has no nontrivial invariant subspace, $V$ is called irreducible (not reducible).
Explanation
This looks similar to the linear-algebra fact that the characteristic polynomial of a linear transformation factors into the characteristic polynomials of the restriction maps on invariant subspaces. In fact, it is essentially the same statement. For a linear map $T : V \to V$ on the finite-dimensional vector space $V$, suppose $V$ decomposes as the direct sum of $T$-invariant subspaces $W_{i}$.
$$ V = W_{1} \oplus W_{2} \oplus \cdots \oplus W_{k} $$
Then the matrix representation of $T$ has the form of a block diagonal matrix whose blocks are the matrix representations of $T|_{W_{i}}$. (See here.)
$$ [T] = \begin{bmatrix} [T|_{W_{1}}] & O & \cdots & O \\ O & [T|_{W_{2}}] & \cdots & O \\ \vdots & \vdots & \ddots & \vdots \\ O & O & \cdots & [T|_{W_{k}}] \end{bmatrix} $$
Similarly, a representation being reducible means it can be expressed in block-diagonal form. For example, if $W_{1}, W_{2}$ is invariant under the representation $\rho : G \to \operatorname{GL}(n, \mathbb{R})$, and $\mathbb{R}^{n} = W_{1} \oplus W_{2}$, then $\rho(g)$ can be written as the following block-diagonal matrix. (We use the same notation for a linear transformation and its matrix representation.)
$$ \rho(g) = \begin{bmatrix} \rho(g)|_{W_{1}} & O \\ O & \rho(g)|_{W_{2}} \end{bmatrix} $$
In other words, the representation $\rho$ can be written as the direct sum of the two representations $\rho|_{W_{1}}$ and $\rho|_{W_{2}}$.