What were the three most important problems Andrew Jackson faced as president of the United States? How successful was he at addressing those problems?

Andrew Jackson's presidency was from 1829 to 1837, and historians generally regard the Jacksonian Era as an extremely important period in the nation's history. The three greatest challenges President Jackson faced were the Eaton Affair, Indian Removal, and Nullification.
Secretary of War John Eaton was not popular with many of Jackson's supporters. Eaton's marriage to Peggy, daughter of a Washington innkeeper, was criticized by society. Peggy, who had a somewhat bad reputation, married Jackson after the death of her first husband. The Eatons were ostracized by most leading families. Jackson, remembering the recent slanderous attacks on his own wife, strongly supported Eaton. Jackson's visceral reaction to the scandal inflamed it and damaged his relationship with John C. Calhoun, the vice president.
A second problem for Jackson's presidency was the removal of Indians beyond the Mississippi River. Georgia was determined to carry it out, and Jackson supported it. Removal was both harsh and dangerous, and many Indians died along the way. Most tribes reluctantly agreed to removal, but the Seminoles bravely fought against it. To this day, many Indians despise Jackson for the Indian removal, which is also remembered as the Trail of Tears.
The third biggest problem for Jackson was South Carolina. That state opposed high tariffs and threatened to nullify federal laws. Calhoun supported the doctrine of nullification. South Carolina's relationship with Washington became confrontational. Jackson deftly managed the situation, avoiding civil war. He threatened South Carolina with the Force Bill but also agreed to the Compromise Tariff of 1833. The situation was defused, but Jackson was prescient enough to realize that further conflicts between South Carolina and the national government were likely.

The Tariff of 1828. This hugely controversial measure, passed during the dying days of the John Quincy Adams Administration, was strongly resisted by the Southern states. The imposition of taxes on imported goods had a particularly negative impact on the predominantly agrarian economy of the South. There was a general expectation that once Jackson was elected, he would take steps to reduce the tariff. When that didn't happen, Jackson's vice president, John C. Calhoun, resigned in anger, precipitating the Nullification Crisis (see below).
Jackson attempted to deal with the crisis by making modest reductions to the tariff in 1832, but the new measures were far too little, far too late to assuage mounting Southern anger and frustration.
The Nullification Crisis. South Carolinians, one of whom was John C. Calhoun, were so enraged by the tariffs of 1828 and 1832 that they convened a special convention at which they declared the tariffs unconstitutional and unenforceable in the state. As states do not enjoy the right to void acts of congress, the actions of the convention created a huge, potentially deadly, constitutional crisis. It seemed that South Carolina was on the brink of succession, and the United States was about to descend into civil war.
Although Jackson prepared for the worst, he also took active steps to deal with South Carolina's grievances. He enacted a compromise measure, the Tariff of 1833, which proposed a gradual reduction of tariff levels until they eventually reached the levels of 1816. South Carolina accepted the new proposals and a major constitutional crisis had been averted.
The Second National Bank. Like most Democrats, Jackson was profoundly hostile to the National Bank. He thought it put too much economic power in the hands of the east-coast financial elite, at the expense of the farmers and land-owners—especially those in the South—who formed the backbone of his support. To Jackson, the National Bank was a corrupt institution which lacked constitutional authority, and so he set about preventing its being granted a new charter.
Jackson persistently used his presidential veto to override any attempts at re-chartering the Bank. His stance, though controversial, was hugely popular with his supporters, and in no small measure contributed to his re-election in 1832. Once more secure in office, Jackson effectively destroyed the economic and political power of the National Bank by removing its federal deposits and diverting them to private banks instead.