There is no relation between Buddha (Gautama who after attaining enlightenment has become Buddha) and the fat laughing Buddha.
The Laughing Buddha is based on an eccentric Chinese Ch'an (Zen) monk who lived over 1,000 years ago and has become a significant part of Buddhist and Shinto culture. This monk was known for his benevolent nature, with characteristic features like large protruding stomach and jolly smile. He popularly became "Laughing Buddha." for his hearty smile.
People believe these cute looking saints bring luck and fortune and keep at home and business places.