The "American Dream" was symbolized in the California Gold Rush by the thrill of exploration, finding gold, and striking rich.
The American Dream is essentially the belief that no matter who you are, where you are, or what background you came from, you can find success in America. You can write your future and amount to wonderful things if you push yourself towards it. This can be seen in the California Gold Rush, as people from everywhere and all different backgrounds would move to California in order to pursue what they wanted, which was to get rich.
The California Gold Rush happened between 1848 and 1855 when gold was found in California. This prompted a ton of people from pretty much everywhere, to move to California and try to find gold. Towns were short-lived during this time, as soon as gold ran out in a place, people would abandon everything and simply move elsewhere to search for gold, but they all did this to get rich.