Everyone has an idea, image / symbol, or story about the American Dream, but there are so many that one simple definition or situation is only a start.
But all the parts fit together in some spirit of ambition, achievement, hope—or maybe just a dream!
Since the USA is known as "a nation of immigrants," the most recurrent example of the American Dream is the immigrant story,
in which an immigrant leaves an Old World of limits and traditions behind
to journey to a land of opportunity and change.
The immigrant's journey never really ends but relocates to new settings or stories: a new city, a better neighborhood, retirement to the exurbs, socio-economic mobility, one's children doing better than oneself.
Other, more general images:
white picket fence (image / symbol) > home ownership
ladder of success