Andrew Jackson is credited with saying "I have only two regrets: I didn't shoot Henry Clay and I didn't hang John C. Calhoun." What was the nature of his relationship with both men that made him hate them so much?

John C. Calhoun and Andrew Jackson were uneasy allies at the beginning of Jackson's presidency. Calhoun was John Quincy Adams' VP, though he essentially ran separately as VP, instead of as Adams' running mate. Calhoun won the VP slot handily, while the POTUS spot in 1824 was decided by the House of Representatives.

During Adams' presidency, Calhoun became an outspoken critic of what he considered to be trampling on "states' rights". Calhoun was a leading voice in support of slavery, which colored his views on Adams' pro-federal policies.

In reaction, Calhoun then threw his support behind Andrew Jackson in 1828, and then became his running mate. Jackson won, and Calhoun won a second term as VP, though, uniquely, under a President of a different party than his first term.

But shortly after Andrew Jackson took office, the Petticoat Affair occurred. Andrew Jackson had appointed John Eaton, senator from his home state of Tennessee, to serve as the Secretary of War. But there was a problem. Eaton was married to Peggy (née O'Neale), whose father owned a "hotel" in Washington DC called the Franklin House that had long been a hangout for powerful Washingtonians, including Congressmen and Cabinet members. It was less a hotel, really, than a drinking establishment, and possibly a brothel as well, though in any case, Peggy had been a regular face at the hotel since she was a little girl, having grown up there. By the time she was a teenager, she was privy to a lot of Washington secrets, and was known as a flirt, and so was not exactly viewed by the powers-that-be in DC as a member of high society. In addition, because of her unique position in Washington, she was not one to hold her tongue on political issues. If she felt a certain way, she would express it, having learned to hold her own in these behind-the-scenes conversations.

So when Peggy Eaton's husband John was appointed Secretary of War, all the other Cabinet wives shunned her, and this shunning was organized by John Calhoun's wife Floride. Back when social customs among high society were a lot more formal, the Eatons were not invited to parties, did not receive visits from other Cabinet members or members of Congress, and were not received when they went to pay calls on others.

This angered Andrew Jackson, and he blamed the Calhouns particularly for orchestrating it. He was angry in part because Jackson's wife had also been dragged through the mud not long before. During the 1828 campaign, Rachel Jackson was accused of bigamy, which was technically true. She had been married before Andrew, but her first husband was abusive and she left him, and was told he had secured a divorce. She went ahead and married Andrew, but it was later revealed that the divorce had never been granted, so once it was, she had to get married to Andrew again--this after they'd had their first children.

It had long been a sore point for Andrew, and when it became a political controversy, he thought it was unfair, and he blamed it specifically for his wife's death. She died in December 1828, after Andrew Jackson won the election but before he took office.

And then a few months later, John C. Calhoun and his wife were pulling the same stunt on the Eatons. And it pit nearly all of Jackson's Cabinet against him early in his presidency.

Only a couple of Cabinet members stayed loyal to Jackson through the Petticoat Affair, notably Secretary of State Martin Van Buren, the lone widower among the Cabinet, so he had no wife to join in the fray. He provided cover for Jackson by offering to resign, which allowed Jackson to ask for the resignations of all the anti-Eaton Cabinet members as well. They all resigned, except Calhoun who was VP and had been elected. Jackson then appointed Eaton to be the Minister to Spain, to get him out of Washington, and appointed Van Buren to be Minister to the UK. And then when Jackson ran for re-election, Van Buren became his new running mate. But John C. Calhoun remained Vice-President, to the great annoyance of Jackson, and even opposed Van Buren's UK minister nomination.

There was another sore point between them early in Jackson's presidency: the "Tariff of Abominations". Under J.Q. Adams, a tariff had been passed that was meant to help industrialists, but Calhoun didn't like it because he thought it disproportionately affected Southern slave owners. He worked with Congress to get the language changed in the bill so that it would affect the North just as much as the South, thinking that Adams would then abandon support. But he didn't, and continued to champion the tariff, and it passed.

This was one of the things that soured Calhoun on Adams, so when he ran with Jackson, the repeal of the tariff was high on Calhoun's list. But Jackson surprised Calhoun by throwing his support behind the continuation of the tariff, which infuriated Calhoun. Calhoun then started speaking out publicly against Jackson's policies whenever one came up that he didn't agree with.

This tariff then led to the Nullification Crisis of 1832. After an additional tariff was passed, South Carolina stated they would not honor either one, relying on a "state's rights" argument that they could nullify any federal law that they pleased. Calhoun supported South Carolina's position, while Jackson did not and took bold action. He got approval from Congress to essentially dissolve the customs houses in Charleston that were run by the South Carolina state government, and replaced them with federal appointees to take over customs of foreign imports at the port of Charleston.

By this time, Calhoun had finally resigned his seat as VP, so that he could run for Senate where he could better champion nullification. The issue was finally resolved uneasily in 1833 when Calhoun (now in the Senate) and Whig leader Henry Clay were able to pass a new tariff that would replace the older tariffs, and the new tariff would placate the concerns of the Southern states. But from a longer view of U.S. history, this put off the question of "state's rights" for another 25 years, a contentious issue that contributed to the breakdown of North/South relations that culminated in the Civil War.

By the end of Andrew Jackson's presidency in 1837, he and Calhoun were essentially political rivals, despite both being members of the Democratic Party and both being Southerners. This, in spite of Jackson's rather pro-slavery position, but a position that wasn't good enough for John C. Calhoun. Calhoun was probably the most vehement pro-slavery politician of his time, and maybe in all of U.S. history. At the end of Jackson's presidency, Calhoun gave a famous pro-slavery speech on the floor of Congress, calling slavery "a positive good". This showed where Calhoun's faction of the Democratic Party stood as the incoming Democratic President, Martin Van Buren, a New Yorker, was already seen as wobbly on the slavery issue. (Van Buren was personally opposed to slavery, but was willing to compromise on the issue, out of fear the issue was going to split the U.S. in half. After his presidency, he became vocally anti-slavery, beginning with a failed bid in 1848 to regain the Democratic nomination for President, and published the Barburner Manifesto, where he stated that the Founders had been against slavery and that slavery should not spread to the territories as those territories were being divvied up as new states.)

There was one last butting of the heads between Andrew Jackson and John C. Calhoun. Jackson had long been in favor of term limits for the President, and spoke about the issue in most of his State of the Union speeches, favoring a single term of six years for the presidency. He had promised to keep the tradition of limiting his tenure to two terms, as many of his predecessors had, but as 1836 dawned, a rumor began to spread that Jackson would be open to the idea of a third term "only" if war broke out with France.

Calhoun took the rumor seriously, and used his influence in Washington to put the kibosh on any third term for Jackson. Calhoun suggested Jackson was planning on starting a war with France in order to hold onto power. Of course, this didn't pan out, and the rumor that Jackson would possibly accept a third term may have been just that: a rumor. In any case, it was an effort by Calhoun to tarnish Jackson's image and paint him as a tyrant, which Jackson did not at all appreciate.

TL;DR: John C. Calhoun was one of the leading pro-slavery voices of his era, and strictly adhered to the social order of high society in Washington D.C. These positions led to the Petticoat Affair, orchestrated by Calhoun's wife, and, indirectly, to the Nullification Crisis of 1832-33, where Calhoun championed South Carolina's position over the Jackson-led federal government. Calhoun and Jackson became bitter political rivals, despite both being members of the same political party, leading Andrew Jackson to state that he regretted he'd never got a chance to hang Calhoun.

I'll leave it to someone else to go into detail about Henry Clay.

Andrew Jackson by William Graham Sumner

The Petticoat Affair: Manners, Mutiny, and Sex in Andrew Jackson's White House by John F. Marszalek

A Critical Study of Nullification in South Carolina by David Franklin Houston

The Autobiography of Martin Van Buren by Martin Van Buren

The Papers of John C. Calhoun: 1835-1837 ed. by the South Caroliniana Society