In the context of this question, popular sovereignty refers to the practice of allowing settlers in a United States territory to decide whether or not to allow slavery in that territory.
As the United States expanded into the West in the nineteenth century, the most divisive political question confronting the nation was whether slavery would be allowed to expand into new territories. The question came up in 1820 with the admission of Missouri and again with the annexation of Texas. But it reached a crisis point in 1850, when the United States gained California and the entire Southwest as a result of the Mexican War. A major political compromise authored by Henry Clay and ushered through Congress by Stephen Douglas admitted California as a free state and established a stricter fugitive slave law. It also mandated that the issue of slavery in the territories of New Mexico and Utah would be decided by popular sovereignty, i.e. by the inhabitants of the territories.
The issue came to the fore again in the Kansas-Nebraska Act, when Douglas introduced a bill to organize those two territories and to allow the issue of slavery to be decided by popular sovereignty. This was very controversial because it overturned the previous Missouri Compromise, which outlawed slavery in those territories. It also led to a bitter partisan struggle known as "Bleeding Kansas," as pro-slavery forces attempted to seize control of the territory.