Slavery in Africa was widespread before the Atlantic slave as well as after the Atlantic slave trade. There's some apologism (interestingly enough, quite similar to Southern U.S. slavery apologism) claiming that it wasn't that bad, but if you look at the actual accounts it could be extremely brutal. Like with the U.S. there was certainly a degree of different experiences, but like in the U.S., that doesn't justify the practice.
In the end it was actually European powers that ended most slavery in Africa, often with a great deal of local opposition ("The End of Slavery in Africa" is a decent starting place if you want to see how it happened in each individual area).
Ethiopia is an interesting example - it wasn't colonized[1], and so slavery there persisted long after it ended in most of the continent. The League of Nations kept pressuring the country to end the practice, but it kept dragging it's feet. It only ended when Italy invaded in the run-up to WWII (it's also interesting as a non-colonized control country when it comes to colonization).
[1] It was conquered by fascist powers for some years, the same as most of Europe.