This is by no means an exhaustive list, but does provide a general overview of the differences between the fish sexes. If you need help with a specific fish, send me an e-mail and I'll give you details for that species.
Angelfish are extremely difficult to sex accurately, particularly when they are young. Occasionally fully mature males will show a modest nuchal hump, which is a bump on the head just above the eyes. However, don't count on it being there in every case.
The best way to establish a mating pair is to purchase a half dozen immature angelfish and raise them together. When they are mature enough they will pair off, and you will have at least one breeding pair out of the group. Once they begin mating, it will be quite obvious which one is male and which is female, as the female will be the one with the ovipositor laying the eggs
Bettas are a species of fish that is generally quite easy to sex. Males have the long flowing fins and brilliant colors that owners find attractive. Male Bettas are what is usually sold in shops.
Females are not as vividly colored and have short stubbier fins. It is not always easy to find female Bettas for sale in pet shops. If you can't locate a female Betta, ask the shop owner or manager if they can order one for you.
Generally speaking, catfish sexes cannot be distinguished. Many species of catfish have not been breed in captivity. The notable exception is the Corydoras species, which has often been breed in captivity.
Cichlids are such a diverse group, that it would take a small novel to give specifics for sexing each species. While many are not easily differentiated, there are a few rules of thumb that apply to quite a few Cichlid species.
Males are often slimmer, but larger than females, and are more vibrantly colored. The dorsal and anal fins of the male are more pointed, larger, and more flowing than the female. In many species the male will display egg shaped markings on the anal fin known as egg spots. Some males have a bump on the head, referred to as a nuchal hump. Although females can also develop a nuchal hump when spawning, it is never as prominent as that of the male. Typically the dominant male will have a larger nuchal hump than other males.
Although the above general rules apply to many species of cichlids, if you are considering breeding them I strongly recommend doing your homework on the specific species before seeking a breeding pair.
Barbs and other members of the cyprinid family are rather difficult to sex. Differences will vary by species, but generally males are more intensely colored and slimmer than females. Because most cyprinids are schooling fish, one way to obtain a breeding pair is to purchase a group of them.
Gouramis are another species of fish that are not easily sexed. Males and females often are similarly colored and shaped. However, there is one fairly universal sexual difference seen in most Gourami species. The dorsal fin is long and and comes to a distinct point in males, while females have a shorter rounded dorsal fin.
In addition to the dorsal fin differences, certain species of Gourami show color variations between the sexes. The male Pearl Gourami has a deep red-orange coloration on the throat and breast. The male Moonlight Gourami has orange to red coloration of the pelvic fins.
Among the easiest of all fish to sex are the livebearing fish. Males are usually smaller and more colorful than the females. They also possess an external sexual organ, the gonopodium, which makes it easy to differentiate males from the females.
The gonopodium is a modified anal fin which is used to fertilize eggs. In the male the anal fin is rod shaped, while the female has a traditional fan shaped anal fin (see photo above).
Tetras are not easily sexed, but do have some differences, which vary based on the species. Generally the females are a bit larger and plumper than males. Males are often more vibrantly colored and may have longer fins than their female counterparts. Because Tetras are schooling fish, breeding pairs can be obtained simply by purchasing a small school of them at one time.
Hopefully this helps a bit. When in doubt, research the specific species you are interested in, and feel free to e-mail me with questions you may have.