Fish species are incredibly diverse, and their reproductive methods are too. Most will breed happily with no intervention, and if there are no predators around, and ample food most should grow up into strong adult fish. Most freshwater species lay eggs, though some give birth to live young (livebearers). The parental care shown varies enoumously. Characins scatter their eggs, and eat any they find, also eating any fry, whereas some cichlids will guard their eggs, and fry in their mouths for weeks. At least one species of catfish cleverly tricks cichlids into mouthbrooding its eggs. Similar to the cuckoo in the bird world.
A lot of fish have never been bred in aquaria, however many have. Here are the main fish breeding methods
Marbles For egg scatterers. This is a great way to breed tetras, danios and other egg scatterers.
Spawning mops For plant spawners. This is a great way to breed killifish and rainbowfish.
Spawning cave For cave spawners.
Peat substrate For substrate spawners. This is a great way to breed annual killifish.
Long term Fry grow up with the adults. Typically used for livebearers and less predatory egg layers.
Avoid tankbusters! Too many tropical fish stores sell small fish that will continue growing until too big for any home aquarium. Rehousing can be very difficult indeed. Avoid species such as the shovel-nosed and red tailed catfish