How was popular sovereignty supposed to work?
In addition to the good information in the above post, "popular sovereignty" was part of the compromise hammered out in 1850, largely by Senator Henry Clay, which allowed Texas in as a slave state, and Maine as a free state. This maintained the balance in Congress between free and slave states.
The westward movement and Manifest Destiny had thrown a monkey wrench in the balance, and the old Mason-Dixon dividing line between new free and slave states wouldn't work anymore after the Mexican-American War. Almost all of the new states from the Mexican Cession would end up being slave states that way.
So popular sovereignty imagined that once a territory reached 50,000 residents and qualified for statehood, an election could be held where the majority rule of the new state could decide its slavery status. This seems democratic, and respectful of the culture and opinions of the territories that were to enter.
Its authors did not foresee the large influx of radicals from both pro and anti-slave forces that would bring so much violence to Kansas.
From a theoretical standpoint, popular sovereignty is a process whereby individuals have a voice through the political process. The idea of this voice is through political participation in the form of voting through direct elections. Essentially, popular sovereignty means that people vote and have a voice through this voting process. Whenever citizens are unhappy with government or pleased with the direction of their governing bodies, popular sovereignty is the way in which those feelings can be voiced. When the framers of the United States Constitution suggested that the goals of the democratic framework, the idea of being able to “ensure domestic tranquility” comes out of the use of popular sovereignty. Changes in government can be articulated through popular sovereignty and not through violence or other similar means. I think that the challenge with popular sovereignty in the modern setting is that it is not truly embraced as a means to voice one’s opinion about government, as seen with low voter turnout in elections.
I assume that you are talking about the term "popular sovereignty" in the context of slavery. If so, here is how it was supposed to work:
What popular sovereignty meant was that each territory was going to be able to pick whether it wanted to have slaves or have slavery be against the law (when it became a state). This was in contrast to how things were under the Missouri Compromise, for example. Under that law, the Congress decided which territories and states would allow slavery and which would not. So this new system would allow the people of the territories to choose rather than having Congress choose for them.