Azura is a deadra. Azura and Meridia are the only daedric princes that aren't really evil - they aren't good either but their actions usually benefit mortals rather than not.
Azura got angry at the 3 guys from the tribunal temple for breaking their oath. Beforehand they swore to Azura that they would guard the heart of Lorkhan and the tools of Kagrenac that the dwemer used on the heart, and that they would never use the tools for themselves, but eventually they did it anyway. Azura was furious and cursed the entire dunmer race for the actions of the 3, with grey skin and red eyes.
It's not mentioned why the aedra didn't punish the dunmer for their own sacrilege and only Azura, who is a daedra, got angry. The deadra traditionally don't care about the actions mortals take against their makers, the aedra, and even encourage such actions. Therefore it's a mystery why the aedra didn't punish the dunmer and only Azura even cared.