Jay Gatsby is one of the main characters of the novel "The Great Gatsby". He is a rich man, who throws awesome parties at his own giant house in West Egg, Long Island. He is in love with Daisy, the woman of his life, which is already married to Tom Buchanan, an aristocrat. Jay Gatsby was born poor, but he didn't want to lead a life like that: he wanted more, he was inspired by the "American Dream". So basically he starts earning money in order to follow his dreams. But that American Dream we mentioned before, was corrupted by the desire of materialism. And so Gatsby starts earning his fortune in an illegitimate way, for example, he sold alcohol during Prohibition.
Jay Gatsby can be considered an antihero because he is corrupted and dishonest. He avoids and makes other people avoid the truth by throwing parties. He hopes to see Daisy again, he hopes to catch Daisy's attention through his extravagant parties, because he desires to love and be loved again. And by doing this, he is only trying to remake the past, which is something that can't be done. He is guided by illusions, and nothing more. This is why he's an antihero: he's corrupted, he follows illusions, not the American Dream, which is already fallen.