"We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness."—The Declaration of Independence (July 4, 1776)
Just like Eagle Land, the American Dream comes in two flavors:
Flavor 1: "Positive"
This came from the perception that immigrants had of America, since they were fleeing from bad situations and places that were hostile to their groups, and finding a place they could begin anew.
The "American Dream" refers to the betterment of position (either for oneself or for one's posterity) that can be achieved in the United States (or in earlier times the more broad "New World" of the Americas). Historically this has meant the obtaining of property. Prior to the discovery of the Americas, the holding of property differentiated between the haves, and the have-nots. Furthermore, not owning property made it extremely difficult to improve one's position in life.
Then the discovery of the Americas meant that there was a ton of land out there to which nobody had a legal claim (the Indians didn't count). If you went to America you could gain ownership of land- and so vastly improve your status in life. Whether this was the yeomen farmers of New England, or the plantation owners of Virgina, suddenly in one generation you could leap from landless renters to property holders- allowing the 2nd generation to pursue things like law and politics (in Latin America things went south because land ownership was not as widely distributed—partially due to the differences between Iberians and the English).
In a single word, the American Dream was: "Independence". No longer would surviving in life depend on your connections to the powerful few, but rather by one's own Diligence, Hard Work, and a bit of Luck. See By Your Own Bootstraps.
As the Industrial Revolution changed the importance of ownership of farm land, the American Dream morphed to include elements such as owning your own small business, or your own home with a bit of lawn. All elements of economic independence in the modern age. This independence meant that obtainers of the American Dream could see their children free to become whatever they wanted to.
Immediately after World War II, the American Dream was the standard idea of a husband, who went to work in a major city, usually by driving to the train station (or being driven by his wife) and commuting; a ranch-style house in the suburbs; two cars and a stay-at-home wife and mom who raised their average brood of 2.5 children.
In modern day America, this tends to mean any number of things, including having a stable job, owning a paid-for house and car(s), and raising a family.
In fiction the American Dream tends to be depicted as either getting really, really rich or else living a happy suburban life like 1950s sitcom characters. The results are commonly either a Rags to Riches story, or Happily Married.
Flavor 2: "Cynical"
This view tends to argue that the American Dream is either not real, being really just a dream, or flawed (or dead). Often complaining that the illusion of some future prosperity is used to keep the "masses" happy while they are oppressed in the present. Those who have read The Great Gatsby will know what this view is like.
In this setting the average citizens are shown to all be busy working on their own Get Rich Quick Scheme, which they are convinced will one day, maybe even tomorrow, make them very rich, and they are, if not uninterested, then outright opposed to the conditions of the losers in life. If fortune waits just around the corner, why worry about such things? After all, losers are losers for a reason. These more cynical takes may also feature the observation that the road to success is a lot more difficult to travel than the Get Rich Quick Schemers anticipate, and that their own search for an easy route to success is holding them back and preventing them from accomplishing anything meaningful; in essence, they themselves become the losers they dismiss.
More recent interpretations reflect on the isolation of Suburbia or the high amount of personal and national debt Americans have accrued over the last thirty years. Families in these works appear to have the perfect life, but it's all a sham. To maintain the facade, the characters may have given up their ideals to lick their soulless boss' boots, or have a secret second life selling drugs, or kill somebody (including themselves) for the life insurance. In this version, you can have the American Dream™, but it comes at a great price, literally and figuratively.
Cynical takes on the American Dream also tend to treat it more as a fantasy, or even a delusion, than a realistic dream. Instead of a land of unlimited possibilities and opportunity, America is a land of limited resources and people all competing for them; this means that there's only going to be so much of the pie available to share, and not everyone is going to get an equal slice. Thus, there's at least one loser (if not more) for every winner; someone's got to clean the streets and scrub the toilets after all.
Again in fiction the American Dream tends to be depicted as either to getting really, really rich or else living a happy suburban life like 1950s sitcom characters. But this time if you go for the former you'll discover that it's Lonely At the Top and if you go for the latter you'll find yourself condemned to an Awful Wedded Life. Take your pick.
Seriously, what the American Dream entails is constantly up for debate, but it's usually agreed that it involves all citizens being allowed to achieve what they want, or at least the opportunity to try. Often it is pointed out that different minorities have commonly been handicapped in chasing the American Dream compared to other Americans. The term American Dream is often used in the context of immigrants coming over to the "Land of Opportunity" (think of the song "America" from West Side Story). Of course, not everyone comes over to America, so a whole lot of people stay with whatever the dream is in their country.
May have some relation to the Ankh-Morporkian Dream, reportedly that of "making boatloads of cash in a place where your death was not likely to be a matter of public policy".
Not to be confused with "The American Dream" Dusty Rhodes, with the Distaff Counterpart/successor/possible daughter of Captain America (comics), Shannon Carter or with the erstwhile TV show American Dreams.