that is a pretty ambiguous term. It really depends on what time frame you are dealing with... are you talking about what was it when the terms were coined? What is it now? What was it during the founding of the United States? the 50's?
The more recent you get, the more people will disagree as to what it means and some re-interpret it to mean what they wish their country was like instead of what the country actually wants itself to be.
My understanding of the American dream is that you can rise from the gutter and achieve wealth, power and happiness. Or alternatively leave school with no decent qualifications and become President. That kind of thing.
Much like the old idea in England of going to London where the streets are paved with gold, etc, etc, to make your way in life.
America is the 'land of opportunity'. This opportunity is the basis of the American dream, AFAIK.
I think the term "American Dream" refers to the alleged ability of a person to make what they want of their life in the US. This is as opposed to many societies where a person is pretty much stuck with what is presented to them. America is reputed to be a place where the opportunity exists to go from "rags to riches" with hard work, and where "any boy can grow up to be president" that latter probably refering to someone like Lincoln who was born to a poor modest family, but rose to the presidency. These notions are sort of true but the reality is vastly more complicated, and the opposite is becoming less and less true in other societies around the world.
The 'American way of life', as uttered by the President in various nationalism-rousing speeches (or speech impediments), is, I believe, a reference to democracy and the belief in freedom, truth, justice, equality, and all the other things that look good on paper but no government actually practises. It is an ideal that people believe in whether or not it actually exists, and thus don't have to ask themselves if it does.
A more cynical interpretation would be to consume and destroy, the MO of most western civilisations. As a mantra, it is far less appealing.
A Japanese guy moved into the building here a few years ago with a second hand cadillac that burned oil and couldn't be smogged or registered till it was fixed. It sat there for about a year, getting dirty. He drove some other small car to get to the university and back. I asked him one day why he'd ever bought the cadillac. "Oh, you know," he said, "the American Dream."