The Rat King is the king of a large society of mice and rats in the sewers. He is lazy, eats a lot, and tries to find a "Suitable wife" which ends in him getting yelled at because, he's a rat.
Whenever Mike and company travel through the sewers for whatever reason, they have to deal with the Rat King and/or his followers.
What he really hates is someone "being a bigger rat than him". When people mistaken Albert as a giant rat, this made The Rat King explode into a rage and sent his army of mice after both Albert and the people who said that.