There is no argument that newly-hatched baby brine shrimp (Artemia) is one of the best things you can feed African cichlid fry, as well as adults in many species, such as shell-dwellers. Of course the lazy man's alternatives to live baby brine shrimp include the more expensive frozen, flake and freeze-dried... but let's say you're both lazy and thrifty, but want your fish to have the best...
There are about as many different methods for hatching and feeding baby brine as there are aquarists, and they range from what I'm going to suggest, to the most technically advanced methods that yield the highest hatching rates. Those that fall somewhere in between include those cute little hatchery kits you can buy at the pet store... IMO, too complicated, and you've defeated your purpose by spending so much on it.
First, you need an area that's out of sight. I've NEVER seen an attractive brine shrimp hatchery set up. I don't recommend setting it up in the living room and then waiting to see if your spouse comes unglued. Find a warm place with an electrical outlet that isn't frequently visited by people that will be annoyed by a light being on. Also needed: empty 3 liter pop bottle, air pump, tubing, clothes pin and an incandescent desk lamp (or reflector, or any incandescent light source that can safely be placed very close to the bottle)... you'll also need artemia cysts (brine shrimp eggs) and uniodized salt (pickling salt is fine, so is aquarium salt).
Fill the bottle about 2/3 full with warm dechlorinated water. Add about 3 tablespoons of the salt and set up the air pump/air line. Poke the end of the airline tubing into the bottle so that it is nearly on the bottom. Use the clothespin to secure the tubing to the top of the bottle. Don't worry about how fast the water is circulating... what is important is that it DOES circulate. Turn it on, make sure all is going well, then add about a teaspoon of eggs. Swirl the water around to get any that have stuck to the side, and put the light on it, very close to the bottle. This will warm the water so that the shrimp hatch. In 24 hours the water will look pink, indicating that they have hatched.
The hatch will be good for a couple of days... to harvest, i like to simply pour some water off into a container (like a plastic bowl) and let it settle, shining a light on the bottom. The empty cysts will go to the top, and the hatchlings will migrate to the light... when you see a pink cluster, use a turkey baster to siphon them up.
What you do next depends on what you're feeding. If you're feeding something that doesn't tolerate salt, you'll need to rinse the bbs in a brine shrimp net or coffee filter. If you're feeding African cichlids, you can probably just put the shrimp directly into the tank without any trouble at all... that's what I do :)
I get the best hatch rate if the temperature is between 80° and 82°F. If you're not getting a good hatch, you might tweak the settings a bit... more or less salt, more or less wattage on the light, or more lights... if it's still not working, your eggs are probably old. I've never had it not work though... GOOD LUCK!