I would say that there are two major ways to characterize what kind of president Jackson was.
First, he was an imperious president who used the power of the office more broadly than other presidents had. Jackson, for example, vetoed more bills than all previous presidents put together had vetoed. Jackson saw himself as the "tribune of the people," feeling that he was meant to protect them. This meant that he felt that he, personally, should wield more power than presidents traditionally had.
Second, he was a president who felt that he should protect the "common man." Jackson felt that his job was to protect the common people from rich and powerful elites. This can be seen in how vigorously he fought against the Bank of the United States.
These are the main ways to characterize Jackson as president.