How did slavery come to an end in various countries?

You ask about the end of slavery in "various countries." I will focus on the British Empire and the US.
Slavery was outlawed throughout the British Empire in 1833, following many years of action on the part of abolitionist politicians and the slaves themselves, whose organized revolts—the first in 1816 in Barbados—indicated to slave owners that the slaves were a powerful group. By the time slavery was finally banned, it was almost universally opposed among the general public, after decades of campaigning by church groups and other bodies. However, until those in political power who supported slavery—largely because they profited from it—could be defeated in the House of Commons by a majority of abolitionists, the bill to outlaw it could not be passed.
Opposition to slavery grew as the slave trade itself became less profitable and less necessary. As the Industrial Revolution progressed, the British cotton and wool industries became extremely profitable and were far easier options for large companies than trading in sugar from the West Indies. The changing public attitude also made it detrimental to the reputation of a company to be a known user of slave labor.
In the US, there were also abolitionist groups, but Americans in favor of emancipation faced a powerful opponent in the form of the Southern planters, who relied heavily upon slaves to pick their cotton, among other things. The US, as a relatively young nation, needed its cotton to help grow its economy: cotton was "king" in the second quarter of the nineteenth century in the US, and the concept of the cotton trade without slaves was difficult to reconcile. While the actual importation of new slaves was banned in 1808, an outright ban on slavery would not become a realistic possibility for decades to come.
After slavery was banned in the British Empire, the power of abolitionist lobbies in the US grew. Now they had an excellent moral argument supported by respected government bodies, but some churches, in fact, attempted to justify the continuance of slavery on religious grounds—that is, because the slaves "needed" to be looked after by whites. It took a civil war to end slavery in the United States, with the Confederacy breaking away upon the election of Abraham Lincoln, a known abolitionist who had sworn to put an end to slavery. These states' dependence upon slavery drove them to secede from the United States, beginning the bloodiest war in its history.
Lincoln finally issued the Emancipation Proclamation in 1863, freeing slaves. Many free blacks fought as soldiers in the war themselves, in defense of their own freedom.
http://abolition.e2bn.org/slavery.html