I have heard some cat owners speak of "belling the cat". What is this, and why would anyone do it to their cat?
It depends on how the term is used I think. I know the term from one of Aesop's fables, where a group of mice gathered together to figure out a solution to keep from being eaten by a cat. They decided to put a bell on the cat so that they'd hear it coming, and be able to run away from it in time. But, when it came time to get volunteers, no one would do it (the usual mumbling and excuses about why they couldn't do it but someone else should).
In this context "belling the cat" is an idiom to agree to an impossible task (or just one that no one will actually participate in). I think it was used in medieval times, referencing when people wanted to overthrow a ruler. I've never heard it actually being used myself.
If I had to guess a more modern use, I'd guess it's taken a more literal meaning, where people are talking about putting a bell on their cat so that they know where it is in their house. That way they know where it is, and if it's causing trouble. That's just a guess though.