Andrew Jackson was one of the most powerful, influential, and controversial Presidents in American history. He was such a remarkable figure that the period from the late 1820s to the early 1840s is often called the "Age of Jackson." Jackson rose to power amidst the expansion of the American electorate to include almost all white males, and he claimed that the office of the President should be used to carry out the will of the American people.

Jackson was a fierce political fighter, and he, along with his second Vice President Martin van Buren, a very cagey politician, purged government offices of political rivals, replacing them with allies loyal to him. As a lifelong Indian fighter, he implemented a policy of removal, forcing the Native American peoples of the Southeast to move to designated Indian territory in modern Oklahoma. This opened up vast expanses of land for cotton growing and slavery. He even ignored a Supreme Court decision that nullified part of the Indian Removal Act, an unprecedented act by a President. As a champion of the so-called "common man," Jackson vetoed a rechartering of the Second Bank of the United States, seen by some as a vehicle for corruption and privilege. This was itself an enormous and fairly rare exercise of executive power, as was his threat to invade and subdue South Carolina when its leaders threatened secession over the Nullification Crisis in 1832-33. Jackson would go on to strangle the National Bank by removing federal deposits from it.

In short, Jackson was very much an activist president, more so than perhaps any previous leader. His political opponents formed a new party, and the name they chose--the "Whigs"--was telling. English Whigs were associated with opposition to royal despotism. Jackson was no king, but he was a remarkably powerful president for the times. 

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.