In The Great Gatsby we see the American dream in sort of a perverted way. Gatsby is chasing the American dream -- he is chasing riches and he is chasing Daisy, who kind of symbolizes the dream. But I say it's perverted because what he's chasing is an illusion -- it can't really make him happy.
In Of Mice and Men Lennie and George are chasing the American dream too. It is represented by the farm they always talk about. They are chasing the dream of being independent and self-sufficient. We don't know if their dream could have made them happy -- society keeps them from finding out.
Concerning your question about the American Dream in the works you mention, The Great Gatsby and Of Mice and Men, you need look no further than the main characters of each.
Jay Gatsby is the embodiment of the American Dream. He makes good, so to speak. Born into a relatively poor family, he achieves wealth, apparently, by hard work and entrepreneurship. He is a self-made man. He owns a mansion and has the finest shirts Daisy has ever seen.
In contrast, George and Lennie are seeking the American Dream, but will never achieve it. Their attempts to raise a stake, as they say, never amount to much. They would like to own their own place and raise rabbits, but it will never happen.
Gatsby achieves the American Dream, but the dream proves hollow. George and Lennie dream about wealth, but will never achieve it.
I read the answers above, but I would just like to have my own point of view for The Great Gatsby. Personally I think Jay Gatsby's real dream was to chase Daisy; he only wanted wealth to impress Daisy.
The American dream is in The Great Gatsby for me is to achieve status, wealth, or anything that is a want for people.
The American dream is very clear in the book Of Mice and Men. George and Lenny talk about their dream of having their own little piece of land on which to farm and raise animals. The idea of property ownership is the symbolist ideal of the American Dream.
In the book The Great Gatsby the American Dream is taken up a notch. Gatsby wanted the dream of what the wealthy had in America. Achieving wealth and living the good lie is also a component of the American Dream but on a larger scale.