What physical system can one use as an example while considering irreps of SU(N)? What is the correspondence between the system and the irreps?
An irreducible representation of a compact Lie group such as $SU(N)$ may always be understood as being the quantization of a certain 1-dimensional topological gauge field theory. In that the representation space is the Hilbert space of states of the 1d TFT and the group action on that is the action of the quantum observables on the states.
In mathematics this statement is known as The orbit method. In physics this statement was famously highlighted in Edward's Witten article on Chern-Simons theory and the Jones polynomial (p. 22, 23).
Details on this story and pointers to references are on the nLab here: The orbit method. See in particular the section Wilson loops and 1d Chern-Simons σ-models with target the coadjoint orbit.