1- Introduction
Definition. Let $G$ be a group and $\mathbb{F}$ a field. Let $V$ be a vector space over $\mathbb{F}$. Then a homomorphism $$ T:G\longrightarrow GL(V) $$ is called a representation of $G$. The dimension of $T$ is the dimension of the vector space $V$.
The image forms a vector space under composition.
Direct sum
Assume $T:G\to GL(V)$ and $S:G\to GL(V)$ are representations over $\mathbb{F}$. Then define $T\oplus S:G\to GL(V\oplus U)$ via $$ (T\oplus S)(g)(v,u)\coloneqq (T(g)v,S(g)u),\quad\forall g\in G,\ v\in V,\ u\in U. $$
Irreducible representations
Definition. Assume $T:G\to GL(V)$ is a representation. A subspace $U\subset V$ is called invariant with respect to $T$ if $$ T(g)u\in U,\quad \forall g\in G,\ u\in U. $$ We can construct the representation $S\to GL(U)$ setting $S(g)$ to be the restriction of the automorphism $T(g)$ to the subspace $U$. $S$ is called a subrepresentation of the representation $T$.
Definition. Assume $T:G\to GL(V)$ is a representatioin. $T$ is called irreducible if it has no non-trivial invariant subspaces.
Definition. Assume $T:G\to GL(V)$ is a representation. $T$ is called completely reducible if for any invariant subspace $U$ there exists another invariant subspace $W$ such that $V=U\oplus W$.
It is an open question whether every representation is completely reducible, but the situation has been proven in the affirmative in the finite dimensional case.