The compromise of 1850 was precipitated by the political crisis that emerged over California's admission to the Union. While Californians (and Northerners) were overwhelmingly in favor of admission as a free state, Southern politicians sought to open it to slavery. At issue, really, was the question of whether slavery should be allowed in the rest of the Mexican cession.
The debate was resolved by a series of compromise measures, proposed initially by Henry Clay. Stephen Douglas guided the measures through the Senate one by one, which was the only way he could get them passed, so intense was the sectional rivalry.
First, California was admitted as a free state. This appealed, obviously to the North. Second, the question of slavery would be decided by popular sovereignty (i.e. by a vote) in the territories of Utah and New Mexico, basically the bulk of the rest of the Mexican Cession. Third, Texas's disputed borders were resolved. This issue had implications for Texas as well as the balance of power in Congress, as it limited the number of representatives from slaveholding Texas. Fourth, the slave trade (but crucially, not slavery itself) was outlawed in Washington, DC, which was a sop to the many petitions for outlawing slavery in the capital that had poured in for decades. Finally, and most galling to the North, Congress passed a stricter fugitive slave law that made it highly illegal to assist a fugitive slave, and gave local law enforcement and judges a financial incentive to hunt down fugitive slaves.
The fugitive slave provision would be arguably the most significant measure, although major problems ensued when Douglas attempted to apply the principle of popular sovereignty to the newly organized territory of Kansas.