Let's see... first the disclaimer: I'm not an actual historian. I was an African Studies minor in college, but I dropped out before earning a degree, and that was 25 years ago. Most of this is from memory, but it's reasonably close to accurate. If you need sources for the Mali section below, hit me and I'll try to dig them up. The Swahili bit is covered deeper in my link. Hope this helps!
There's the Egyptian Empire, which at times covered a good portion some of what's now called the middle east, up to and including pretty much the entire Levant region at its height. A lot of folks don't know that the Egyptians spread that far north and east.
The Mali Empire is, to me, the most fascinating. From circa 1200 to 1600AD, founded by Sundiata Keita (who is mythologized in a great story cycle that's been published in English several times - search for his first name and you'll find it). It covered a large swath of west Africa, from the Atlantic coast to the inland Niger delta. What's so fascinating about the history here is that this region transitioned from traditional local religion to Islam during the course of the history of the nation, as well as the early days of European colonialism. The kings converted pretty early, and even in the later days of the nation there were plenty of non-Muslim citizens. The empire was fairly well documented in the Arabic world, especially after Mansa Musa made his Hajj with an enormous entourage. He was insanely rich, and threw around enough gold that it cause major economic shocks in Cairo. Ibn Battuta, an Islamic explorer and scholar, was so intrigued that he visited and became part of the royal court for some time. His reviews from an Arabic perspective were mixed - local customs and attire bothered him a lot (slaves, even women, were often naked) but he admired the zeal of the Islamic believers. Timbuktu - one of the three main global Islamic learning centers in the medieval period - was founded as a city during this era.
Another not very well known (and the one I spent most of the time studying) is what's called the "Swahili Coast," which was never really a full "empire," but a series of trading outposts up and down the east coast of Africa. Centered in the Zanzibar Islands, and primarily ruled by a sultan established in Kilwa (an island off the coast of what's now Tanzania), this was a huge trading organization that shipped goods from inland African peoples as far east as India. The sultan 1300s? ish when it was established, was of Omani extraction originally. I gave a pretty extensive history on this one here: