The Nullification Crisis began in November of 1832, when South Carolina passed an Ordinance of Nullification, declaring the tariffs of 1828 and 1832 null void, prohibiting the collection of the tariffs after February 1, 1833. After South Carolina nullified the tariff laws, Vice President John C. Calhoun resigned his office and then was chosen to represent South Carolina in the U.S. Senate. Andrew Jackson sought increased power from Congress to enforce the tariff law. In January of 1833 the Force Bill was introduced in Congress that would give the president the power to collect customs duties in South Carolina using the full power of the government if necessary, meaning the use of military power. This bill went nowhere until a compromise tariff bill was introduced in Congress by Henry Clay which provided for the gradual lowering of the tariff until 1842 when no duty could exceed 20%. Both bills passed Congress and on March 2, 1833, President Jackson signed both. With the lowering of the tariff and the threat of military action against South Carolina, and with no other southern state backing South Carolina, on March 15, 1833, South Carolina repealed its nullification ordinance. The nullification crisis came to an end, though 3 days later, to preserve what it felt was the state’s right, it nullified the Force Act. Though the crisis came to an end, the debate over the right of a state to nullify federal laws continued.
Basically, the nullification crisis was handled because Andrew Jackson acted with a combination of toughness and gentleness.
Andrew Jackson was a Southerner and so his attitude mattered a lot. When he came out strongly in favor of the national government (and against nullification) it was very important. He acted very toughly by speaking out against nullification and even threatening military action.
At the same time, though, he was more conciliatory. He worked to get Congress to reduce the tariff that made the South Carolinians so angry.
So Jackson's combination of toughness and kindness settled the crisis by getting South Carolina to back down.