The slave trade is extremely old and has involved numerous states. Those who participated in the slave trade and then banned it did so at different times for different reasons. The British spearheaded the ban of the trade in the early 19th century and had swayed most of the major players to ban the trade by 1820 (America independently banned the trade around the same time as the British, 1807-1808.

But the European colonial powers weren't the only ones to engage in the slave trade. The Barbary Coast was notorious for its participation in the slave trade, though galley slavery at least held the hope of eventual release (prisoner swaps and ransoms were common; Cervantes, of Don Quixote fame, was a galley slave at one point). We also saw the Knights of St John engage in the galley slave business.

Then there were the slavers of East Africa, roughly from modern Sudan to modern Mozambique. The Atlantic trade had no real impact here. The British started anti slave trade campaigns in the region in the late 19th century, but that didn't impact the more northerly parts of that regional space trade -- the influx of slaves to the Ottoman Empire didn't really care about British ships off Zanzibar. Oddly, the East Africa slave trade thrived after the various bans of the early 19th century.

The slavers of West Africa were more adversely affected. While some ships would still buy slaves and try to run the British blockade, the space trade was rather badly hurt by the British efforts and self-imposed abstention on America's part.

But all the European bans and patrols in the world couldn't end the practice. The slave trade only existed because slavery in Africa was exceptionally widespread by the time the Europeans showed interest. And slavery would often remain part of daily life in African states until change was forced upon them. That doesn't mean the slave trade didn't impact African politics: it most certainly did. Myriad African petty wars were waged for the purpose of acquiring slaves.

As you can tell, this is a very large topic.