Finite diagrams
A finite diagram has no ultraviolet or infrared divergences — the loop integrals converge at both high and low momenta, giving a well-defined numerical result without regularization.
Physical meaning
Finiteness can be enforced by symmetry. The light-by-light box, for example, is UV finite because gauge invariance forces the amplitude to carry enough powers of external momentum in the numerator to tame the high-energy behavior. Tree-level diagrams are always finite because they involve no loop integrals at all.
Resolution
No divergences to resolve — the result is directly physical.