The cat that he was holding in "The Godfather" was a stray that he found. He insisted that he hold it in order to show both the hard and soft side of his character. Apparently the cat was purring so loud that it drowned out the voices of the other actors and they had to do voice overs for that scene.
He also had a pet raccoon named Russell that his Mom gave him. He used to take it everywhere and Russell used to follow him around like a shadow.
Here is another angle for the picture posted earlier. I don't know the name of his kitty though.
I never knew why it seemed so taboo for us men to hold cats but now after a few years on this site, I think I've finally figured it out: we make you ladies jealous
Obviously Brando didn't buy into the stereotype. But then again no one would ever disrespect the godfather anyways. Clearly it takes a real man to love a cat.