What compromises did Congress propose in dealing with slavery in the territories and how well did it work?

The main ones were the Three-Fifths Compromise, the Missouri Compromise, the Compromise of 1820 and the Compromise of 1850. None of them worked very well and the Civil War broke out in spite of these compromises.

In the Three-Fifths Compromise, the South wanted to count slaves for tax purposes but the North was opposed because it would give the South more representation in the House of Representatives, since those Congressmen are alloted by population. Slaves could not vote, so the slave owner would vote for the slaves, making this an unfair situation to the North. This compromise declared that a slave would represent 3/5ths of a white person or free black person for purposes of population. Dumb idea, huh?

The Compromise of 1820 said that there had to be an equal number of slave states and free states in the Union. Each time a territory wanted to join the union, there was a big fight over whether the state would be free or slave. In 1820, there were 11 slave states and 11 free, but Missouri wanted to join the Union, but it was a slave state. Luckily, Maine also wanted to join, so Maine came in as a free state and Missouri came in as a slave state. This was known as The Missouri Compromise, but this didn't last long.

The Compromise of 1820 worked for awhile, but 30 years later, some other states wanted to join the Union (California, Texas, etc.) so this compromise allowed California to enter as free, Texas to be divided up and enter as a slave state, but the most important provision of this compromise was the Fugitive Slave Act. This provision made it illegal to harbour escaped slaves. Northerners were required by law to return any escaped slaves.

There were a lot of harsh provisions in the Compromise of 1850, but it worked for awhile. Eventually, the South wanted to secede from the Union over slavery, but other issues as well - tariffs, taxes, sectional and cultural differences, etc.