Uncle Billy is everybody's weird uncle. Weird in a positive, absentminded way, not weird in a "we suspect him of plotting against the government" way. He's an amusing eccentric—a likable character but a bad businessman. His main weakness is that he's totally forgetful. He needs to keep strings tied to his fingers to help him remember certain things he's supposed to do. Not a great trait, considering that he's George's co-partner in running the Building and Loan (which Billy founded with George's dad). Also, on the likable/lovable side of the score, he's apparently an animal lover; he has a pet crow and a pet squirrel.
Billy sparks the main crisis of the movie when he misplaces $8,000. This almost ruins the Building and Loan and makes George seriously contemplate suicide. OTOH, you could argue that, without Uncle Billy's screw-up, George never would have had his epiphany about how wonderful his life is. Uncle Billy's role in the movie is to help him reach that point.
When George sees what the world would be like if he'd never been born, he discovers that Uncle Billy would've wound up in an insane asylum, having lost the Building and Loan to Mr. Potter. This implies that, if the Potter-type people were in control, the lovable eccentrics like Uncle Billy would all be doomed. They require a protector and defender, someone compassionate to look out for them.