Historical BackgroundThe Whig Party, in the United States, was for most of its history concerned with promoting internal improvements, such as roads, canals, railroads, deepening of rivers, etc. This was of interest to many Westerners in this period, isolated as they were and in need of markets. Abraham Lincoln was a Whig for most of this period.
The name came into use in the 1680s in England when there was the threat of establishment of a line of Catholic Kings, starting with James II. The Protestant element, who held that Parliament could prevent such a succession, came to be called Whigs after a radical Presbyterian group in Scotland, the Whigamores, while the party tending to the doctrine of the rights of King James II (and naturally containing Catholic as well as simply royalist elements), were called Tories after some bands of Irish Catholics who had been driven to become outlaws due to the crusade of the English against the church they clung to.
The designation of British loyalists during the American Revolution - as Tories - is well known. And many on the revolutionary side must have identified with the English Whigs, which continued to be the party in favor of Parliament's keeping the king in check.
During Andrew Jackson's presidency the first really well organized political parties came into existence. The Democratic Party, with Jackson himself as the rallying point, brought about radical changes, including a presidency that for the first time threatened to overshadow Congress.
Jackson changed the perception of the presidential veto. It had been interpreted as something the president could do if he considered a bill unconstitutional, though the Constitution doesn't really state it that way. Jackson eventually laid down the precedent that the president could veto a bill on basically any grounds.
Jackson also took a free hand in changing the membership of his cabinet in a way that no other President had done. He had his first cabinet resign en masse, which to many at the time looked like the "fall of the government", but he treated it as something a President had the right to do.
All of this lead the opposition, to speak of King Andrew I, and to picture Jackson that way in political cartoons.