There were actually two political parties in the United States that disappeared during the 1850s. These were the Whig Party and the Free Soil Party. Of the two, the much more important was the Whig Party. Therefore, that is probably the answer that you are being asked to give.
The Free Soil Party was a relatively small party. It was only in existence long enough to run candidates in two presidential elections, those of 1848 and 1852. Its main platform centered around preventing the spread of slavery into the territories.
The Whig Party was a much bigger party. It emerged during the 1830s as a party that was mainly opposed to Andrew Jackson. It was one of the two major political parties from that time through the 1852 presidential election. However, between 1852 and 1856 it essentially disappeared. It took about 44% of the popular vote in 1852 but did not even run a candidate of its own in 1856. (It endorsed the Know-Nothing candidate.) The Whigs split apart because the Northern Whigs were opposed to slavery while the Southern Whigs were in favor.
Thus, while both the Free Soil Party and the Whigs disappeared in the 1850s, the Whigs were the more important party. Both of these parties’ supporters generally moved to support the new Republican Party.