Often, in quantum mechanics I found the sentence "The Hilbert space carries a representation of $SU(2)$ group" (in particular when dealing with anglar momenta).
Effectively, I know that this means that the eigenvectors of some operators ${({\bf J})^2}$ span the Hilbert space and they can be a base for the vector space. Where $\bf{(J)^2}$ is the Casimir ${\bf (J)^2}=J_x^2+J_y^2+J_z^2$ and ${\bf J}$ belongs to the algebra of $su(2)$, namely e.g. $[J_x,J_y]=J_z$ etc.
But what does it means in general in terms of representation theory?