Being that there are literally thousands of different species of African cichlids, there is no way I can tell you about every possible combination. Instead, what I can do is give you some generalities. Not only will you be more inclined to stay awake, but this will also be more helpful. These generalities are based upon my experience and research. There are four main areas of discussion that are important to address if we are to come to a general understanding of cichlid compabilities. These are their diet, water requirements, aggression, and cross-breeding tendencies.
Simplistically speaking, there are three main classes of cichlids: vegetarians, meat-eaters, and those who depend upon a combination of both. The meat eaters, or predators, have high animal protein requirements. These are typically the Haps, or open water fish that grow to be rather large. The vegetable eaters are the mbuna, who primarily subsist upon algae. There are, however, many fishes that have more of a balance between these two groups. These are the peacocks, utaka, and Tanganyikan shell-dwellers. While they do eat algae, they also depend heavily upon invertebrates and insects.
Having said all of this, mixing these different groups can be done if fed a quality, "well-rounded" food. There are many quality commercially-prepared foods that will meet the nutritional requirements of all three groups. If mixing these different types of dietary groups is in your plans, the most important consideration is ensuring that you don't mix predators with smaller cichlids, as the former will dine on the latter.
Some cichlids prefer to gather their food off the bottom of the aquarium, while others readily consume it at the surface of the water. Realizing this, many manufacturers now offer pellets that sink and others that float. By mixing floating and sinking pellets, you can keep the bottom-grazers (e.g., peacocks, frontosa) and surface feeders (e.g., Utaka, Mbuna, Protomelas spp.) content. Additionally, mixing pellet sizes can also keep fishes of different sizes happy. Some manufacturers have even developed pellet foods with a variety of sizes in a single container.
Another important difference between some African cichlids is their differing water requirements. This is much simpler to address and categorize than any other difference. Cichlids from the Rift Lakes should not be mixed with fishes coming from the rivers and deltas surrounding these lakes, such as Pelvicachromis pulcher (a.k.a. Kribensis). These cichlids are known as the riverine cichlids. They inhabit freshwater streams with a very neutral pH of 7.0, whereas the water in Lake Malawi and Lake Victoria have a pH between 7.4 and 8.6. Also, the Lake Tanganyika cichlids typically should not mixed with those from either Lake Malawi, Lake Victoria, or the rivers surrounding these lakes as it has a pH between 7.8 and 9.0. Tanganyikans will survive and can do quite well, but you will need to keep the pH towards their lower end of tolearability.
The take-home message is that it's best not to mix cichlids from different lakes with the exception of those from Lakes Malawi and Victoria.
One of the most problematic issues surrounding cichlid compatibility is that of aggression. Cichlids display the most aggression towards others of the same species and other conspecifics (or fish with a similar morphology). This is because fish with a similar appearance are most probably closely-related. And if they are closely-related, then they probably have the same dietary requirements and are a threat to a polygamous male's harem, as cichlids of different species can cross-breed. (More on this below). So, if they are similar in appearance, they are perceived as a threat.
In order to cut down on the aggression and mortalities in your cichlid tank, aviod putting fishes of the same genus and/or color together. For example, avoid putting two different Victorian Hap species in the same tank, as they all have very similar coloring, barring, and shape. Electric blue is a very common color for many Lake Malawi cichlids. Sciaenochromis fryrei, for example, are not good tank mates for many of the Aulonocara species, many of whom are dominated by a similar ice blue color.
Mortality and morbidity are not the only reasons you shouldn't keep cichlids of similar shape and coloring. I have had species of differnet genera, but of similar coloring who proved to be problematic. Let me explain. I have a very dominant Hap. sp. 44 "Red Tail" that has prevented my Pundamilia nyererei from ever successfully spawning. Everytime they have tried to spawn, he will dart across the tank and position himself right in between them! This went on for a year and finally I removed the the nyererei to their own tank where they are much happier.
So, the lesson here is do not mix cichlids of similar shape or coloring.
Another reason why you should not keep two different species from the same genus in your tank is because of cross-breeding. As a general rule, fish from the same genus are subject to cross-breeding. Take for example these two females pictured below: Aulonocara saulosi on the left and Aulonocara hansbaenschi on the right. They are a typical representation of the Aulonocara species females, which are all almost completely indistinguishable. While the Aulonocara spp. females are a dramatic case, they are representative of this problem among all cichlid genera.
This can really become a problem if you keep only a male of one species and the female of another. And believe it or not, if you keep males and females of both, you can still get cross-breeding. This is because the male of one of the species will become "super dominant" and may in fact annex the female of the subdominant male.
Why is cross-breeding bad, you may ask. Well, for starters, hybrids can cause problems in the aquarium because their behavior may be unpredictable as far as compatibility is concerned. Furthermore, cross-hybrdization (even between geographic races of the same species) dilutes gene pools, producing progeny that may be sterile and/or lack the original color of either of the two parents. If you accidentally cross a Haplochromis sp. "Ruby Green" with a Haplochromis sp. "Flameback," which look very similar and could be innocently done, your fry will not have the brilliant red stripe of the Flameback or the emerald green of the "Ruby Green." You have then swindled a buyer if you ever plan to sell them. It doesn't matter if your buyer knows what he's getting because imagine what happens once they leave his hands, or those of the person to whom he gives those fish (or their progeny). Many of these fish, especially the Victorian Haps, are now disappearing from their natural habitats and it is up to us, the hobbyists, to keep these cichlids alive and preserve them in their original form.
So...don't keep cichlids of the same genus, and especially not the male of one and only the female of another.
Other Tank Mates
A common question is “What other kinds of fish can I put with my cichlids?” Well, the response to this question depends HEAVILY on which Africans you're talking about. In general, fish that do well with Africans are Synodontis catfish, most Botia Loaches, most Labeo Sharks (including related fish such as the “Flying Fox”), larger Rainbowfish & Danios, most suckermouth catfish (Plecos) and the larger Tetras (the “African Redy-Eye Tetra,” “Diamond” Tetras, “Black Tail” Tetras, and “Congo” Tetras. Be aware that some of these have neutral pH requirements but can tolerate higher pHs, like those required by African cichlids. This is something to take into consideration when mixing cichlids with other freshwater fish.