The main historical significance of Andrew Jackson is that he was the first president to come from a common background and to embody a populist vision of politics. He is well-known for having been suspicious of elites and for having pursued policies that, to him, weakened the elites and strengthened the common people.
Perhaps the best example of one of these policies was Jackson's "Bank War." He felt that the Bank of the United States was a tool of the rich elites. He felt that it let them keep control of the country's economy and let them act in their own interests and against the interests of the common people. Because of this he fought to kill the Bank.
After Jackson, this sort of populism became more common. It continues to this day and can be seen quite clearly in "Tea Party" attitudes towards such things as the Federal Reserve.