Fish breed in many ways, and yes you can watch. In fact, watching fish breed is one of the great fascinations in the hobby because there are so many interesting breeding strategies among fish.
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.