 Solution

A relevant operator has $y_i>0$, so its coupling grows under coarse-graining and drives the system away from the critical renormalization-group fixed point. Such directions must be tuned to reach the critical manifold. An irrelevant operator has $y_i<0$, so its coupling decays and normally supplies corrections to scaling. For $y_i=0$ the operator is marginal; nonlinear terms in the transformation decide whether its coupling grows or decreases, or whether it labels a line of fixed points.

This classification is a statement about perturbations of a specified fixed point, not an intrinsic label independent of the theory. For example, a quartic scalar interaction has eigenvalue $4-D$ at the Gaussian fixed point, so it is relevant for $D<4$ and marginal at $D=4$. A tricritical scalar point has an extra relevant even direction compared with an ordinary critical point. Although irrelevant couplings usually disappear from leading powers, a dangerously irrelevant coupling can still be needed to stabilize an ordered phase and invalidate naive hyperscaling.