Renee was Moe's girlfriend.
She initially only went out with him because she felt sorry for Moe that he couldn't get anyone. In spite of that beginning, though, they were a happy couple for a long time. Renee broke up with Moe after she found out that Moe wouldn't bail Homer out of jail, instead choosing to spend money on a trip to Hawaii for the two of them. Renee was shocked by Moe's callous treatment of his friend. When Moe started to tell her about a plan he had (involving corpses and burning down the bar) to both bail out Homer and take the trip, she became disgusted and left.