You don't give a time scale or any particular sub-region / exclusion, so I assume you mean the whole continent. (I'm considering Kush/Kerma/Napata/Meroë to be "Egyptian" enough to avoid special mention, though I love those upper Nile societies myself.) There are an enormous number of distinct states and societies we might call civilizations, although the term civilization is pretty nebulous when you try to define its edges. I'll page /u/Commustar on this too, as his flair is specifically on the Swahili Coast and Sudanic Kingdoms, which were remarkable.

Out of the very many, I'll focus on a few. The successive complexes in the far south (Mapungubwe and the 'Leopard's Kopje' cultures just north of the Limpopo; the proto-Shona state we know as Great Zimbabwe; and the Mutapa and Butua Empires, the latter of which became the Rozwi) are very noteworthy because they built monumental structures. Mutapa really didn't, but the others were in areas of workable stone and so built with it. Innocent Pikirayi's now-dated The Zimbabwe Culture is one of the key works, but the journal Azania has been a recent showcase for new work on the Zimbabwe plateau. Chris Ehret's supposed to be coming out with a new edition of his pre-1800 volume (Civilizations of Africa: A History to 1800) which, while textbooky, is actually quite thought-provoking.

Another I like to point towards is Asante (or Ashanti, as it's sometimes known). As one of many states in the West African forests, it's noteworthy for its cohesion and its status as a net importer for most of the era of the Atlantic Slave Trade. Although it formed around 1700, it was based on smaller city-state domains in the Akan Forest which had been loosely aligned or at least politically connected a thousand years prior. The emergence of Asante itself is really a crucial moment, because by the late 1700s it became a true bureaucratic state with a national identity--arguably, it was a nation-state in every sense, except without writing. There's a lot of work on Asante out there; Ivor Wilks talks about the knowledge and culture structures, Sara Berry and Jean Allman have done crucial work on Asante society, and others like Joseph Adjaye have written on key aspects like diplomacy. Wilks is probably the most comprehensive though quite academic, but a general survey volume or bigger history like Toyin Falola's would be good too.

I suppose I have to mention the Kingdom of Kongo and the great deal of work John Thornton has done on it as well--it was a powerful, centralized, militarized state that unfortunately fell victim to court intrigue abetted by Portuguese ambitions in west central Africa. As a state, it had remarkable organizational structures, aimed at providing for the power and stability of the state. Again, John Thornton's your person there; he and Linda Heywood have written a great deal.

This just scratches the surface. I could list a great many. I'd suggest looking at a recent general text (published in the last 3-4 years) to see how it's all interlinked and find out more. Although there is still a shortage of work on these societies as historical entities compared to Europe, the US, and much of Asia, it is out there.

If you'd like to watch documentaries, all of the episodes of BBC4's Lost Kingdoms of Africa series and PBS' Wonders of the African World are available on YouTube.

The various kingdoms of Ethiopia are pretty interesting examples of African civilization. The Kingdom of Beta Israel in particular is really odd--it was a Jewish (!) kingdom in eastern Africa which existed up until the 17th century. They were militarily powerful at times and at one point conquered the Christian kingdom of Aksum. I'd love to give you some sources on them, but they're a bit hard to find--if anyone knows somewhere where we could read more I'd appreciate a link.

Aksum was also an interesting one, but since you're specifically asking for non-Egyptian civilizations it is a bit tricky. Aksum was a Christian Kingdom whose ruling family claimed descent from Solomon. The Christian Kingdoms in Ethiopia were influenced by Coptic Christianity as it arose in Egypt, though, so it might fall under the scope of Egyptian civilization by some definitions, but it really is quite different and it is probably doing a disservice to the Ethiopian Christians to lump them in with Egyptian Christian communities.

[ You can read more about Aksum here ] ( http://www.dskmariam.org/artsandlitreature/litreature/pdf/aksum.pdf )

Don't know if I can give you much to work with material-wise as I haven't been able to find any for myself, but Western Africa had 2 very notable empires : the Mali Empire and the Songhai Empire. Both very rich from trade (gold, salt and slaves mostly if I'm not mistaken). The Mali Empire eventually collapsed with some help by their 'successor' state, the Songhai Empire, which was a vassal state of the Mali. The Songhai would fall to the Moroccan gunpowder troops but still live on in a diminished state until the colonial powers arrived to Africa.

North of those, you could look into the Berber States of North Africa, which eventually became protectorates of the Ottomans and fought the European powers of the Mediterranean for supremacy of the sea. They were mostly known for their piracy, including the well-known corsair Barbarossa who was named Bey (governor) of Al-Djazair.

The Ancient era had Carthage, which is very well-known. And there's Ethiopia which has a rich history.

The continent is very large, so there's a lot more, and like it's been said, there's no time of reference.

Piggyback question. Do we have any experts on the Malian empire on this sub? Every few months I post the same question about Musa I, and it always gets upvotes but no one ever responds.