The terms describe how the coupling terms of the theory change as one increases the energy. A theory is renormalizable = critical if the coupling terms remain unchanged, super-renormalizable = sub-critical if the coupling terms decrease with increasing energy, and non-renormalizable = super-critical if the coupling terms increase (and eventually diverge in the limit that the energy tends to infinity).
These lecture notes by Witten describe the classification in some detail, with examples, from the perspective of perturbation theory (Feynman diagrams). For a presentation in terms of PDE's (stochastic heat equation), see Stochastic PDEs, regularity structures, and interacting particle systems by Chandra & Weber.