It should be pointed out that the Election of 1824 was one of two decided by the House of Representatives, as no one had a clear majority in the Electoral College, although Andrew Jackson did hold a plurality of the votes. Jackson's opponents in that election were John Quincy Adams, Henry Clay, and John C. Calhoun. Calhoun withdrew from the race and ran unopposed for Vice President, leaving three men in contention. As noted above, Clay also withdrew and threw his support behind Adams, and was appointed Adams' Secretary of State. It was not the prestige of the office of Secretary that infuriated Jackson; it was the fact that the office of Secretary of State was considered a stepping stone to the Presidency. Several previous presidents (Jefferson, Madison, Monroe and Adams) had all served as Secretaries of State; therefore it appeared to Jackson that Adams and Clay had virtually locked him out of the Presidency for anywhere from eight to sixteen years. It should be noted that although the "corrupt bargain" position played well with the public and was used effectively by Jackson and Martin Van Buren, such a "bargain" was unlikely. John Quincy Adams was a principled man who hated politics and political dealing. It would be completely out of character for him to make such an arrangement. Also, Henry Clay had a visceral hatred of Andrew Jackson and opposed him at every turn, particularly while Jackson was President.
It was not so much Jackson's supporters who were infuriated as it was Jackson himself. He began his campaign for the 1828 election almost immediately after the election of 1824 was elected. Although electors were still chosen by state legislators, Jackson carried his campaign to the people in the first truly modern electoral campaign. Speeches, barbecues, fireworks, etc. kept his name before the public (and through them the electors) such that he defeated Adams handily in the 1828 election.
The Corrupt Bargain was a deal that was allegedly made to decide the presidential election of 1824. In this bargain, it is said that Henry Clay gave the votes of his supporters to John Quincy Adams. In return, Adams appointed Clay to be his Secretary of State, which was a more prestigious position in those days than it is now. This bargain allowed Adams to win the election even though he won less of the vote than Andrew Jackson did. This bargain infuriated the people who supported Jackson and helped him win the election of 1828. The Corrupt Bargain is seen as an example of the bad old way of politics in which elites controlled outcomes. Jackson's win in 1828 is seen as an example of democratization with the "little people" taking power away from the elites.
Corrupt bargain was a factor in the election of John Quincy Adams. During the election of 1824 between William Crawford, John Adams, Henry Clay, and Andrew Jackson, Jackson had won the most popular votes and electoral votes. However since none of them had won a majority, the election was decided by the House of Representatives according to the 12th amendment. The speaker of the house was Henry Clay, and he gave his support to John Quincy Adams because Adams supported the American System as well. But not only that in return, Adams gave Clay the job of Secretary of State. Jackson and others were furious and referred to this as the "corrupt bargain".