Unfortunately I am on my tablet and thus unable to give a proper detailed response without going absolutely mad with that awful tiny virtual keyboard here. But I'll try to give the gist.
So Buddhism's main guy is of course its founder, the Siddhartha Gautama. His most common depiction is as a tall and slender young man, sometimes even skeletal in appearance.
The general story is that the young prince Siddhartha, disgusted by the obscene affluence of his life compared to his people, chose to abandon all his worldly possessions in favour of an asketic lifestyle. Even thought he later realised that asketism was not the answer either and chose a middle path it is still common to depict him as basically a sack of bones in some areas.
Now what you're asking for is the jolly laughing guy most people in the west would primarily associate with Buddhism. That's actually a completely different guy called Budai, a Chinese Buddhist monk who lived 400 years after the "original" Buddha but also achieved enlightenment and thus is also technically "a Buddha" (depending on the specific interpretation but generally just a term for someone who achieved enlightenment).
The "laughing Buddha" is a central figure of Chinese Buddhism and thus mostly seen in China while the slender Siddhartha Gautama is found primarily all over India and most of southeast Asia. Although how and when the Chinese version came to be the dominant form in the West I have no clue whatsoever, my only guess would be Chinese immigrants to the US but that's baseless speculation, which is rightfully frowned upon here.
hi! you'll find more on statues of Buddha & Budai in these posts