The main historical significance of Andrew Jackson is that he was the symbol of the democratization of the United States. This is not to say that he caused the democratization or that he, himself, was a perfect role model. It is to say, however, that he personified a major development in American history.
Andrew Jackson was the first president to come from outside the elite stratum of American society. He was, famously, a backwoodsman. He grew up fairly poor and was largely self-educated. This is much different from the backgrounds of people like George Washington and Thomas Jefferson.
Because of his background, Jackson was something of a roughneck. He was well-known for his acts of physical violence. These include his duels and his willingness to execute militia members during Indian wars.
When in office, Jackson was also a partisan of the common (white) man. This is most clearly seen in his struggle against the Second Bank of the United States. Jackson killed the bank largely because he felt that it was a tool of Eastern elites which they used against common people.
Of course, Jackson had his flaws. He was willing to flout the Supreme Court on the issue of Indian Removal, for example, when they ruled Indians could occupy land in the United States but without holding title. But this sort of behavior was common to most people at the time who believed that whites were a superior race.
Jackson was elected president at a time when the country was democratizing. The right to vote was being extended to all white males instead of being limited to those who owned property. There was coming to be the sense that politics was something that all white men could and should participate in. Jackson symbolized that era.