The previous thoughts were very strong. I would like to add one more element. I think that political leaders used compromises that never really solved the slavery issue in a totalizing manner. this is how they dealt with it. There were deferrals, such as the Constitutional Convention's attempt to revisit the slave trade in 20 years. There were localized attempts to take it out of the hands of the federal government, such as popular sovereignty. There were strong denouncing claims that were made in name only. The reality of the matter was that Jefferson was prophetic in his comparison of slavery to a wolf being held by both ears. One cannot let go, but one does not want to hold on to it. In the end, this is how leaders approached the issue of slavery primarily because all other problems such as the political party emergence, the debate of big and small state representation, as well as other issues presented to the new nation could be negotiated away. Slavery was one issue where there could really be no compromise. If slavery was a moral evil, then it had to be denounced and banned. If it was an issue of tradition, then it needed to be upheld and defended. In this light, compromise on slavery might have worked in the short term, but failed in the larger configuration.
Great answer above, and quite complete. I would add only a few thoughts and clarifications.
You can make the argument that the compromises did work pretty well, considering the fact that we were able to forestall the Civil War from 1789 all the way to 1861. The compromises, and our talent for compromising, bought us 72 years of time.
You can also argue that some worked better than others. The Missouri Compromise also included the establishment of the Mason-Dixon Line as the future dividing line between slave and free states, which ended the debate, at least for a while, as to how new territory would be divided. The Compromise of 1850, however, was a miserable failure, leading to Bleeding Kansas in a mere four years with no new states successfully admitted under Popular Sovereignty voting.
You can also say the Fugitive Slave Act failed as a compromise measure designed to pacify the southern interests. It was impossible to enforce, and abolitionists openly violated it. Even the Supreme Court ruling of Dred Scott vs. Sanford in favor of slaveowners' property rights did little to stem the flow of runaway slaves northwards.
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.