In the United States, it was of course the Civil War that led to abolition. However, I assume from your tags that you are asking about slavery in British possessions and/or the Atlantic slave trade.
The British ban on slavery did not come directly out of any dramatic events like the American Civil War. Instead, it came about more gradually. It can be said to have started with Somersett's Case, in which a British court ruled that a slave should be freed and said that there was no basis in British law for slavery to exist. That was in 1772. This made slavery illegal in Britain, but the institution persisted in the colonies.
By 1808, the British had banned the slave trade. This came not out of any single event but more as an outgrowth of the ban on slavery. Finally, in 1833, slavery was banned in all British territories. This was partly because of the natural evolution of thought in England. However, it was also partly due to an uprising in Jamaica in late 1831 known as the Baptist War.