G. Smith already said everything relevant, I try to dive a bit more
into the physical side, which will be only special cases but they cover
more than just one field of physics:
Let's assume we are given a field theory described by the Lagrange Density
$\mathcal{L}$. Now, I assume you are familiar with Noethers Theorem (which also
holds for discrete systems described by a Lagrange Function $L$).
Now, given a symmetry of the system the theorem tells us that there
have to be conserved quantities. In order to understand how representations
are relevant in this context, its enough to understand what exactly a symmetry is. We can assume that there is a Matrix Group $G$
which is acting on the state space (or just the space in which our field
takes it's values), and acting in the natural way, e.g. $g \psi$ is the matrix multiplication pointwise: $g \psi (x) = g \cdot \psi(x)$ for
$g \in G$.
Now let us generalize this! First of all, $g$ might depend on $x$, secondly
we can take any group $G$ (not necesarily a matrix group!) and definie
a representation, call it $\rho$, on $GL(V)$, where we assumed $\psi:
M \to V$, and $M$ is just the space on which $\psi$ lives, so
$M= \mathbb{R}^{1,3}$ in the case of minkowskispace (which we assum in what follows, this does not really matter for the whole picture).
That is (by definition), a group homomorphism
$$\rho: G \to GL(V). $$
Now, being a symmetry group of a system means (not exactly, but for simplicity
let's assume the following, it really does not matter for the whole picture) that $\mathcal{L}$
be invariant under the action of the given group, that is:
$$\mathcal{L}(x,\rho(g)\psi, \partial_{\mu}\rho(g) \psi)
= \mathcal{L}(x,\psi, \partial_{\mu} \psi) \ \forall g \in G$$
where we allow $g$ to depend on $x$.
One can then show, that $\rho(G) \subset GL(V)$ is a Lie Group, so
one typically assumes $G$ itself being a Lie Group and $\rho$ be continous.
Thus, in the conetxt of symmetries of a field theory (which is pretty general I'd say!), representations are needed to understand how the symmetry Group
affects the system,
which in turn is needed to study the conserved quantities corresponding to this symmetry group!
In short we say, that each symmetry gives rise to conserved quantities,
by which really the above is meant, and the way we call it is more or less
a consequence of being lazy about technical details.
Again, this holds for any field theory (classical or quantum) as well as
for discrete systems, so for almost any theory we have!