When someone is beyond angry. Full of rage. Hornets are really angry and relentless, hence the term, "mad as a hornet".
1. Jane was playing hide and seek and was trying to hide in the closet quickly, but smashed her finger in the closet door. Jane was mad as a hornet.
2. If someone is very upset, you ask them "are you mad as a hornet?" Their reaction will tell you how mad they are.