Alright, knowledge learned from a college level class on African History I half attended, go!
To build off/amend the points made by jacob, West Africa had most of what you would classically define as empires. However, some significant points need to be made in understanding how society was structured and why. In general, despite popular belief, Africa is a pretty fertile and naturally wealthy region. The issue has never been owning the land, for the continent is massive, therefore the problem is having the labor to work it. This has wide-reaching implications for societies and states, as the reason why it never crystallized into a patchwork of states like much of the rest of the Old World did, is because of the lack of capability or even necessity in intensive development of the land on the level that other regions required. Not to say that Sub-Saharan African methods of agriculture were unsophisticated: it was just really damn easy, and once set up the amount of effort required to maintain it wasn't as significant as in some of the harder fields in N. Europe, for example. And when the population grew and the capability of the surrounding land maxed out, they just moved out and established new villages. Going back to the issue of labor, this is how power would generally be defined, and thus where the tradition of slavery began in Africa. Though the size of the land meant that slavery was held in a different light than in the American colonies, because if you really mistreated your slaves chances are they'd just run off.
Now then, with that established, let's move back to empires. The issue of famine that jacob mentioned I'm not as familiar with, though if I had to hazard a guess given the dates, it might be related to the climatic changes resulting from the Little Ice Age. Before Sokoto and Bornu though, which fall more into the early modern period in terms of chronology, for the classic West African Sahel empires of Ghana (300CE to 1200CE), Mali (1235CE-1600CE), and Songhai (1464CE-1591CE) the basis of power was more based around exchange and trade. The overlapping dates account for the continuation of prior empires as rump states or as puppets for the sake of legitimacy, btw. The classic example is the trade of salt, from the Sahara desert, in exchange for gold, from mines in the rainforests in the south. However, control over exchange was hardly limited to these items, and control manifested in a few ways, from direct sponsorship of trade by the Mansa, to securing the routes politically and keeping the routes safe by hunting down bandits. I'd postulate that imperial power increased further by the time of Imperial Mali with its espousing of Islam, which in addition to linking it further to the North African states and empires, also reinforced controls over trade through precepts of Sharia law (which, again, is not merely a set of social regulation but had a significant amount of say in mercantile jurisprudence).
To answer the second part of your question, again using the examples of the West African Empires, they were quite advanced, probably on par in some respects, and likely more advanced in yet other respects, than their contemporaries in feudal Europe. The security of trade routes in Mali at its height was legendary, as was its wealth in gold, of course. They also sponsored significant architectural projects and more importantly the construction of universities, where they attracted students and staff from across the Muslim world. They maintained a significant army, though I don't remember if it was a standing army or not.
Now in regards to your last question, it's a bit difficult to say. There are some aspects that seem feudal, but then again they aren't exactly exclusive to a feudal society. Regardless, I'll say that, for the most part, Africa did not really have a structured feudal society. There were hierarchies, oftentimes with aristocrats, though ownership based on heredity varied depending on region and cultures, or even on the particular village. However, in the case of the empires, rule over provinces was given to governors, who in turn were put in place by the Mansa, and with a few exceptions were not hereditary posts. From there, however, government control was fairly loose and organization was still based around city-states and tribes.
Going back to Jacob's points, in East Africa empires were more sparse, but for reasons more complex than the presence of absence of food surpluses. They did, however, exist, but they manifested more as shifting hegemonies between the big mercantile city states of the Swahili coast, whether it is Pate, Zanzibar, Somalia, or eventually Oman.
Central and South Africa had states as well, some of them quite considerable. Central Africa had the Kingdom of Kongo, which acted as an equal in dealings with the Portuguese for centuries, and is a fascinating subject in its own right, as well as the Kingdoms of Ndongo and Matamba, in particular led by the indefatigable Queen Nzinga Mbande. South Africa would for the most part remain stateless, for reasons discussed above, but with the very notable exception of Great Zimbabwe, though it remains quite mysterious as we lack written records for it, as despite its wealth and size its contact with the outside world was handled largely through intermediaries.
I'll end with this, as this was probably one of the main points of the course as it was imparted to me. Africa was not a marginal continent but was an integral part of the world trade system, whether it is that of the Old World or the emergent and truly global network including the New World, and its history is tied up with the greater history of the world, not separate from it.
Oh, and this deals largely with West Africa, though I touched on the other regions, I didn't quite give them justice. The basic principle I started with largely applies to all of them, but they are all extremely interesting on their own, so if you'd like I can write on the other macro regions of Africa and share what I know.
guess N-word posting is back on the menu, then?