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 nearly impossible to sex accurately. 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.
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.
Bettas are a species of fish that is 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.
Females are usuully more robust when viewed from above and are usually bigger.
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 species.
The only sure way to determine the sex of egg laying fish is to do what the experts call "venting" the fish.
You must remove the fish from the water (net it out of the tank) and turn it upside down (hold it very gently, they can be hurt easily) so you can examine the vent holes in the anal area. Only do this over a bucket of tank water so if you drop the fish or it flips out of your hands, it will fall into the water in the bucket.
You will see that the females will have a rounded vent (hole) and the males will have a tear-drop shaped vent. If the fish are young, it will be very difficult to discern the difference in shapes.
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. Some males have a bump on the head, referred to as a nuchal hump.
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.
Cyprinids / Barb Gender
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.
The females however, keep their pale yellow noses (may turn slightly pale red at times, which makes it tricky to determine their sex in store). They only show a small area of red at the tip of the dorsal fins, and ventral fins will stay pale red. In females, their stripes stay black.
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.
Perhaps the easiest 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.
Generally speaking any fish that gives birth to live young is usually easier to sex. These are known as sexually dimorphic. These include the family,Poeciliidae, which are mollies, guppies,swordtails,platies and wags. They are easily distinguishable by the gonopodium which is male modified anal fin used for copulation.
The female has an anal fin shaped like a triangle. As is seen in some of the chidlids, the male has much more distinctive dorsal fins than the female. Even these fish can be difficult to tell until they are sexually mature. Up until then they pretty much look alike.
Telling Koi Carp apart for example is pretty impossible until the female is ready to spawn.
Ultimately the best way to tell the gender of fish is to obtain some information on the species you are interested in. If you study their behavioral differences as well as physiological make up you may be able to make an informed decision. Another source of good information is your local pet store who can probably give many helpful tips.