Intuitive

The trivial representation maps all group elements to the identity matrix. This representation exists for any group.

Spin-$0$ particles and fields transform under the trivial representation of $SU(2)$. That is, their spin value does not change when being rotated.

The action (and often the Lagrangian) transforms under the trivial representation of space-time and gauge symmetry groups. That is, the action is not affected by these symmetry transformations.

Concrete

Example

The diagram below shows the defining representation of $SU(2)$ in its upper branch and the trivial (1-dimensional) representation in its lower branch.

For a more detailed explanation of this diagram see Fun with Symmetry.

Abstract

Why is it interesting?