Jackson's legacy is complex. He was such an influential figure that many historians still use the phrase "Age of Jackson" to describe the United States between the 1820s and early 1840s. This answer will focus on three aspects of his legacy.
The first is that Jackson and his New York political ally Martin Van Buren very shrewdly took advantage of the new, more popular and democratic politics that emerged in the United States during this period. Most states relaxed or eliminated property restrictions for voting during the 1820s and 1830s, and this empowered ordinary white men to vote in elections for the first time. Jackson's appeal to the "common man" and Van Buren's mobilization of working-class people in New York City in particular would permanently change politics in the United States. After Jackson, active campaigning, the dispensation of government jobs, organized political parties, and the affectation of a popular political style all became essential parts of the new mass politics that was, it turned out, there to stay.
The second part of Jackson's legacy is that he expanded the role of the Presidency. While this, again, was largely a matter of style, Jackson was seen as an activist president, someone who believed that he was put into office to carry out what he believed to be the will of the people. His veto of the (re)charter for the Second Bank of the United States, as well as his decision to ignore the Supreme Court's decision in Worcester v. Georgia, contributed to this legacy. So did his decision to take a hard line against the nullifiers in South Carolina, a stance that put him at odds with many Southerners. That the Whig Party that formed largely in opposition to Jackson's policies took a name that recalled opposition to the King in seventeenth and eighteenth century England says much about how Jackson's political enemies viewed his largely unprecedented exercise of executive powers.
The final part of Jackson's legacy is the most shameful one, but it is tied directly to the emergence of a democratic politics in the United States during his presidency. We have to remember that the democracy Jackson and his followers (the so-called "common man") envisioned a white man's democracy, sometimes termed a "herrenvolk democracy" by historians. The most stark example of this is Jackson's support for the removal of the Southeastern Indians. This policy ranked highest on Jackson's agenda upon entering office, and it resulted in the removal of thousands of Cherokee, Creek, Chickasaw, Choctaw, and Seminole Indians from the American Southeast. This process, which we would term "ethnic cleansing" today, also paved the way for the dramatic expansion of slavery into the fertile soils of Alabama and Mississippi, as planters brought thousands, even millions, of slaves to cultivate cotton. So it is important to remember that "Jacksonian democracy" that is so important to Jackson's legacy, and positive portrayals of him in textbooks, was in large part dependent on the taking of Native lands and the expansion of slavery.