In humans, as I'm sure you know, the nose isn't necessarily required for breathing to take place, so one could say that the nose's main role is in olfaction or odor detection. The membranes in the sinus receive and process odors which are then relayed to the brain. Other functions for the nose include humidifying incoming air and initial filtration of large particles by resident hairs. As far as animals breathing via other organs, certain amphibian species can absorb oxygen from water through their skin, rather than respiring through their nostrils. Although this may not meet your requirements for "breathing", it serves the same purpose.