There are two main strategies that fish use: egglaying and livebearing.
Livebearing fish do what the name suggests. The female gives birth to fully formed, free-swimming young. The female fish is internally fertilized by the male fish, and carries the fry for about a month before delivering them. Upon delivery, the babies swim off, hide, and begin searching for food.
Livebearers include the popular mollies, platies, swordtails, and guppies. Other livebearers are halfbeaks, anableps, and fish in the Goodeid family. They are easy to sex, as the female is larger, and the male has a rod-like anal fin called a gonopodium that he uses to internally fertilize the female. After fertilization, the female can produce multiple batches of babies without a male present.
Egglaying is also what the name suggests: the fish lay eggs instead of giving birth to little fish. As the fish grow, they hatch into fry with an attached yolk sac, and then mature into fish. The process usually takes around a week to 10 days, although it can vary widely.
Substrate spawners are a little choosier about where they put the eggs. They lay eggs that attatch to some sort of substrate. Plants, rocks, wood, and even the aquarium glass may be chosen as a spawning site. Both fish participate in the egg laying, with the male fertilizing the eggs as the female lays them. Examples of substrate spawners are many catfish, some cichlids, and killifish.
Bubblenest builders lay their eggs in a nest of bubbles blown by the male fish. The bubbles are held together with saliva and look like foam. They tend to attract infusoria that the babies can eat, and keep the eggs at the surface of the water, where they are well-oxygenated. The eggs are laid a few at a time, and carefully placed in the nest where they hatch. Examples of bubblenest builders are bettas and gouramis.
Mouthbrooders actually keep their eggs in their mouths until the eggs hatch. The eggs are again laid a few at a time, and once the male fertilizes them, the parent doing the mouthbrooding gathers them up in his/her mouth. That parent eats sparingly, if at all, until the baby fish are released. Examples of mouthbrooders are male arrowanas and female cichlids.
Marine fish also lay eggs. Some are substrate spawners, but many lay pelagic eggs that float in the plankton. There the eggs hatch into a larval stage, and the larvae float freely and eat tiny plankton until they grow into fish. See the Moe reference for a more complete description.
Many fish parents show some common behaviors, so I will discuss them here.
Most fish consider any and all fish eggs and young to be a tasty treat. Therefore most fish will not hesitate to snack on any they find, including their own. This means that egg scatters and many substrate spawners really cannot be bred in a community tank, as the eggs will quickly be eaten by the parents and other fish. Marine fish and invertebrates also eat eggs. Livebearers are especially notorious for eating their young.
A few fish ignore their eggs or fry, and so can be bred in a species tank. White cloud minnows can breed this way, and many killifish will at least ignore the eggs. Baby killies are fair game, though. Guppies will also often ignore babies.
Other fish have one parent that guards the eggs and fry. Most bubblenest builders and mouthbrooders operate this way, as do some substrate spawners. The responsible male or female stays with the eggs and young, until they are free swimming. With bubblenest builders, the male tends the nest, blows bubbles as they pop, and keeps any falling eggs or fry in it. He will also defend the nest against other fish. Mouthbrooders simply hide their eggs in their mouths, and some substrate spawning catfish will hide the eggs underneath them. Certain substrate spawning cichlids also have one parent care for the eggs and fry.
A more common setup among cichlids is to have both fish guard and care for the young. This setup can be really fascinating to watch. The parents will take turns fanning or blowing fresh water onto the eggs, and removing any fungused eggs. They will also fiercely defend the spawning site, which can often cause injury or even death to other tankmates. Once the eggs have hatched, the parents will also guard the fry. Some fish will even move the fry to a different place each day. Once the babies are free swimming, some fish continue to guard them, while others end their parental duties. Many African cichlids guard their babies until they spawn again. Discus even feed their babies off of their slimecoats.
A more extreme version of guarding is practiced by some Tanganyikan cichlids. There, older siblings will stay around the nest and help the parents defend subsequent spawns. The babies are allowed to stay until breeding age, when they are driven off.
A breeding tank also is good because it can be kept clean. Eggs and fry need very clean water to hatch and grow. There are also no adults around to compete with the babies for food. Many breeders use a bare tank with only a sponge filter as filtration. Debris and extra food are easily seen and siphoned off daily. Frequent water changes can be done on the tank, as there are no other fish around to stress.
Another solution is to allow fish to breed on yarn mops, a plant, or a piece of slate or glass in the community tank. The eggs can then be moved to the breeding tank to grow. This works well for angelfish, catfish, and Australian rainbowfish. Killifish eggs can be collected from peat or yarn mops and set in a separate container or dried to incubate. Livebearers can be bred in a commercial breeding trap or breeding net within a community tank. The trap separates the babies from the mothers and then gives the babies a safe place to grow.
Some cichlids protect their babies well enough to just be left in a community setup, although this can stress the other fish in the tank. In fact, there are species of cichlids that will turn on each other if there are no other fish in the breeding tank for them to threaten.
Baby livebearers are usually the easiest to raise. Some will take finely crushed flake foods from the start, and only require frequent water changes to keep up with their growth. They also need algae or spirulina.
Baby egglayers are often more difficult to raise. Most are too small to eat adult fish foods, and so require special foods. Live baby brine shrimp are the food of choice for most baby fish, although some require even smaller infusoria. Sifted daphnia also work. Baby algae eating catfish require algae or blanched vegetables. There are also commercial fry foods that work or, in desperate situations, cooked egg yolk. Be careful, though, because non-living foods pollute the tank water terribly -- especially egg yolk.
Actually, keeping the tank water clean is probably the biggest challenge in raising fish. The growing fish require lots of food, and they are not very good at finding it which means even more must be added to the tank. As in any fishtank, adding lots of food must be balanced with keeping the water quality extremely high. In fact, fry require cleaner water than adult fish. Frequent water changes are a must, as is efficient biological filtration. Baby tanks often require daily water changes of up to half the tank. Sponge filters are the preferred method of filtration because they are great biological filters but cannot suck up baby fish.
Marine fish larvae have the strictest requirements of all. They must be fed extremely small plankton or rotifers in a tank with near-perfect water. For more discussion of marine fish rearing, see Moe.
Finally, as the baby fish grow, they must be transferred to larger quarters. Clearly the 10 gallon tank that housed 100 fry cannot house those 100 fish for long. Betta breeders have even more work on their hands, since the little male bettas will fight and have to be put into separate jars or a partitioned tank.
As for turning breeding into a commercial venture, remember the laws of supply and demand. For most common community fish, pet stores can order whatever they want whenever they want it from importers, fish farms, and wholesalers. The hobbyist, on the other hand, has occasional batches of fish that the store may not need or want at that time. The only thing on your side when you walk into a store with a batch of unrequested fish is that locally bred fish are often healthier and less stressed that fish that have been shipped and must be acclimated to local water conditions.
If you insist on breeding saleable fish, try rare catfish, rare rainbows, African cichlids, show quality fancy guppies, or marine fish. Those are all difficult for stores to obtain. To make money selling more common fish like angels, barbs, tetras, cory cats or livebearers (other than guppies), you need many breeding tanks and breeding pairs of fish to assure a constant supply. You must also have fish of consistent quality.
Personally, I would recommend that you breed fish for the sheer pleasure of it, rather than turning your fun hobby into a business venture. There is nothing like seeing a pair of ciclids court, disappear into a cave, and emerge in a few days with a swarm of babies.