In a movie about police snitches and spies, the word "rat" gets thrown around an awful lot. But Frank Costello takes this term to the next level when he develops a psychotic obsession with the idea of a rat in his gang. He starts drawing horrifying pictures of rats and says stuff like:

COSTELLO: Point I'm making here is, Bill, I got this rat—this gnawing, cheese-eating, f***ing rat—and it brings up questions.

In this scene, Jack Nicholson even improvises this amazing rat-like gnawing motion that helps convey just how crazy the thought of a rat is making him.

The longer Costello goes without finding out who his rat is, the angrier he gets. It even comes to the point where he starts threatening the people who are closest to him, for example when he tells Colin:

COSTELLO: I hope I won't have to remind you that if you don't find this cheese-eating rat bastard in your department, you can assume it won't be me who pays for it.

By the end of the movie, we've become so entangled in the world of rats that Martin Scorsese decides to close the show with the image of a rat running across Colin Sullivan's banister. Rats, after all, are creatures that can survive (and even thrive) in very harsh circumstances, and that's definitely been the case with Billy Costigan and Colin Sullivan, even though neither of them makes it out of the movie alive.

We've seen how community is a such a big issue for the people in this movie. There's the Southie community, and then there are the communities of cops and mobsters. Disloyalty is a pretty huge issue in this context—but given the way the social structure in Southie is set up, it's pretty much inevitable. Maybe that's why Frank gets so obsessed: the presence of a rat in his mob is evidence of the fact that really, he can't ever trust anyone.