Andrew Jackson was a war hero who fought in the battle of New Orleans. He was popular with the common man in the election. He lost in controversial fashion, due to the "Corrupt Bargain."
John Quincy Adams was the son of former president John Adams. He was the Secretary of State for James Monroe, before winning the election. The win is tainted though, because of the "Corrupt Bargain" in which he allegedly told Clay that he would give him the job of Secretary of State if he gave him the votes, and Clay did.
Henry Clay was a senator from Kentucky, and the speaker of the house before the election. Clay came in third in the election of 1824, but was a big role in Adams winning. Clay was part of the "Corrupt Bargain". Clay was the Speaker of the house so if he gave his vote to Adams it would be huge, Adams offered him the secretary of state role, which Clay liked. So, Clay became the Secretary of State under Adams.