The fact that many of the new immigrants were Roman Catholic sat poorly with much of the United States's largely-Protestant population. In particular, many Protestants viewed with distrust the strong allegiance of Roman Catholics to the Pope; many Protestants saw this allegiance to the Pope as an allegiance to a foreign prince (especially since the Pope is also head of state of The Vatican), and thus dangerous to American independence and possibly even treasonous.
These concerns spawned widely-held conspiracy theories regarding the Pope's purported plans to subjugate the United States through a continuing influx of his followers. The fact that Popes in the past had in fact wielded significant power and entangled themselves in wars and political disputes was frequently pointed to as evidence that the Pope was simply waiting for the right time to regain his lost temporal power, and served to further cement this notion in the minds of many Americans.
While significant in their own right, the concerns about the Pope largely exacerbated already-present anti-immigrant and anti-Catholic feelings held by many Americans. The newcomers differed culturally from most Americans, so their influx was seen as a threat to maintaining American culture. The immigrants brought with them their strong accents, Irish traditions and culture, and Roman Catholicism, and didn't seem likely to become "Americanized" anytime soon. A vocal minority of American Puritans also protested against the immigrants' alcohol consumption. While alcohol consumption already quite popular among the general public in the United States (even George Washington had run a distillery), the Puritan elements managed to promulgate a caricature of the Irish as drunkards, and thus gained some support for their anti-alcohol crusade by painting drinking as foreign and un-American.
The growing anti-immigrant sentiment led to a dissatisfaction with the major parties -- the Democrats were seen as too dependent on the votes of immigrants, and the Whigs were seen as ineffectual, and were largely in decline in any case. Thus anti-immigrant activists began splitting off from the major parties and forming secret groups, coordinating their votes and throwing their weight en masse behind candidates sympathetic to their cause (regardless of political party). When asked about these secret organizations, members would reply "I know nothing," which led to them popularly being called Know-Nothings. The Know-Nothings in effect gained control of a large number of local offices, especially in the North, through the early 1850s ("in effect" because the officeholders were still technically either Democrats or Republicans, as the Know-Nothings were not yet an actual party).
In 1854 they won significant victories in Congress and at the State level, again as an unofficial party driven by coordinated votes for sympathetic candidates; the secret societies themselves supplemented by supporting votes from the population at large, once it was made known who was sympathetic to the Know-Nothings' cause. The results of this election were so favorable to the Know-Nothings that they formed officially as a political party, called the American Party, and swallowed many members of the now nearly-defunct Whig party, as well as a significant number of Democrats, especially Northern Democrats.
The height of their success came in the Election of 1856, in which they threw their weight behind Millard Fillmore (a Whig who had been president from 1850 to 1852). Fillmore lost, but won 22% of the popular vote and Maryland's 8 electoral votes. However, by this time the newly-formed party was beginning to be rent by differences over slavery, and greatly declined in strength. Know-Nothings remained strong on the local and state levels in many northern states, but by the Election of 1860, they were no longer a serious national political movement. Most of the anti-slavery Know-Nothings went on to join the new Republican Party.
The platform of the American Party called for, among other things:
Causes of the Movement
Distrust of the Pope
Culture Clash
The Political Movement
History
Platform
See also: List of political parties in the United States