George and Lennie's dream is to own a house and a few acres of land. They'll have cows and pigs, a vegetable patch and chickens. They'll have rabbits for Lennie to tend and an alfalfa patch to feed the rabbits. When it rains, they won't have to go to work; they'll stay inside by the fire. They'll have an extra bunk so that if a friend comes by, he can stay.
Their American dream is one shared by many Americans: they want to own their own home and be able to support themselves. They don't hope to be rich, but to get by and make their own decisions. They want to enjoy the simple pleasures in life.
For Lennie, the dream comes down to the rabbits. But for George, this dream will allow him to be his own boss instead of having to go from farm to farm looking for work. It will allow him to care for Lennie in a safe place and possibly to marry and have a family of his own.
Like so many Americans during the Great Depression (and even today), George and Lennie fail to achieve their American dream when George kills Lennie at the end of the book. It's clear from the ending, when George goes off with Slim to get a drink, that the dream is dead and George will spend the rest of his life working for other men.