There is no accuarte representation of what Buddha looked like as there was no recording in his lifetime. I may be wrong about this, but I heard that the fat guy was actually supposed to be a monk at one point and was just coopted into buddhaness.

The fat buddha was a guy named Hotei, aka the happy buddha, who was a wandering beggar monk noted for his jolliness, girth and his street school where he educated poor children. He is said to have always carried a bag of treats for the kids who followed him about.

He came to be a symbol of luck and prosperity and so is often seen in restaurants and businesses, often surrounded by pretend coins, flowers, incense and candy.

It is considered good luck to give his tummy a little rub as you go by.

Here is a traditional koan about him:

Hotei lived in the T'ang dynasty. He had no desire to call himself a Zen master or to gather many disciples about him. Instead he walked the streets with a big sack into which he would put gifts of candy, fruit, or doughnuts. These he would give to children who gathered around him to play. He established a kindergarten of the streets.

Whenever he met a Zen devotee he would extend his hand and say: "Give me one penny." And if anyone asked him to return to a temple to teach, again he would reply: "Give me one penny."

Once as he was about his play-work another Zen master happened along and inquired: "What is the significance of Zen?"
Hotei immediately plopped his sack down on the ground in silent answer.
"Then," asked the other, "what is the actualization of Zen?"
At once Hotei swung the sack over his shoulder and continued on his way.