The idea of a group representation gives flesh and bones to the abstract notion of a group. A set of objects satisfying the group axioms has no tangible meaning, and that is why we need to represent those objects in a way that allows us to work with them.
Consider a Lie group $G$ and a vector space $V$. One can construct the group of general linear transformations on $V$, $GL(V)$. A representation $V$ of $G$ is a Lie group homomorphism, from $G$ to $GL(V)$:\[\rho: G\to GL(V)\] (Note the abuse of terminology here: we call $V$ the representation, not $\rho$.)
In physics textbooks one often reads that some field $\phi$ furnishes a representation of the Lorentz group (which is a Lie group). The point is that we know how the Lorentz group acts on spacetime (on $(t,\mathbf{x})$), i.e. we know one representation of the Lorentz group, and we want to find how it acts on functions on spacetime, i.e. on $\phi(t,\mathbf{x})\equiv \phi(x)$. Of course these functions are the fields of quantum field theory. (Note that we will not consider internal symmetries like $SU(3)_\text{color}$ here, but the discussion extends trivially to those.)
It turns out that the Lorentz group acts on scalar functions (corresponding to scalar fields like the Higgs field), spinorial functions (corresponding to spinor fields like the electron field), vectorial functions (corresponding to gauge fields like the photon), and the list goes on. The quantum number that distinguishes all these representations is spin. Note that the spaces where these fields live are representations of the Lorentz group, and that is where the terminology we mentioned in the beginning of the previous paragraph comes from.
The important point is that the representations of the Lorentz group correspond to fields which, after quantization, correspond to the particles we observe in the real world.
