In F. Scott Fitzgerald's The Great Gatsby, what is Gatsby's real first and last name?

In F. Scott Fitzgerald's The Great Gatsby, we learn that the titular character was born with the name "James Gatz," and known as "Jimmy Gatz" for much of his life, but changed it to Jay Gatsby. 

The reason he changed his name is because he wanted a new identity to reflect his new lifestyle. To him, "Jimmy Gatz" was a lower-class farm boy who had been dumped by the love of his life. However, if the name changed form, to "Jay Gatsby," it could be the name of a millionaire who deserved to win someone like Daisy Buchanan. 

Daisy's cousin and the narrator of the novel, Nick Carraway, believes that that Gatsby had his new name ready for a long time. It came from his "Platonic" representation of himself, meaning that he saw himself in a certain "perfect" way and was conforming however he could to become that ideal self. Instead of James Gatz of some unnamed farm in the middle of nowhere, he becomes Jay Gatsby of West Egg, Long Island, one of the wealthiest neighborhoods in America. "James Gatz" could never live in West Egg.

He changes his name the very moment he sees his first opportunity to make money: helping a man called Dan Cody to smuggle alcohol into American from Canada through Lake Superior. Thus, when he is just seventeen years old, Jay Gatsby earns a fortune through illegal means, and leaves Jimmy Gatz behind forever. 

It is also the effect of Gatsby's wild and vivid imagination. He always saw himself as this type of man: wealthy, refined, respected. Because of this, he never accepted his parents as his own, or his background and heritage as his own. Gatsby's dreams are a reflection of the so-called American Dream in many ways. He had a romanticized vision of his own future, which was at odds with his upbringing. Thus, to become a successful and rich magnate, he had to leave a lot of things behind--including his name. He wanted to be someone else so badly, that he changed his entire identity in order to leave his root behind. His imagination is an important part of his character. It is what keeps him pining for Daisy all those years, and what allows him to finally make a fortune and recreate himself in order to get her back. The name change is simply a reflection of this.