Except in the case of birth defects, fish usually regrow fins.
Causes of Injury
If your fish's fins are split, it's typically a sign of physical damage. This can come from a number of sources. Aggression from other fish is a common source of torn fins. This can take the form of fighting among fish or less serious fin-nipping from smaller fish like barbs. Fish can also injure their fins on aquarium decorations with sharp edges and ridged aquarium plants. Fish sometimes get injured during transport in aquarium nets as well.
Infections can also cause fin damage. If a bacterial infection is the culprit, the fin with typically have a ragged, possible fuzzy edge to it. Additionally, you may see red streaking or slimy mucus on the fish's fins. The bacteria that cause fin infections are typically opportunists, meaning they usually cannot infect a healthy fish, but if the fish is stressed by poor aquarium conditions or has injuries from a fight, infection could strike.
If you see damaged fins, you can take steps to promote healing. Above all, a fish in a tank with clean water will be more healthy and likely to regrow the fin than a fish in a dirty tank. Perform additional 25 percent water changes to remove fish wastes from the water. Normally, you only need to perform regular water changes every two to four weeks, but if you see infection or injury, you should be doing it at least weekly. Some species may benefit from the addition of one tablespoon of aquarium salt per gallon of water. However, look up your aquarium fish; a handful of species, such as cory catfish, cannot handle even tiny doses of salt.
When to Use Antibiotics
In some case, you may need to use antibiotics to allow your fish to regrow fins. Usually, a fish will regrow fins without incident. But sometimes, fin rot and other disease may impair its ability to do so. If you see signs of infection and salt doesn't help, consider purchasing antibiotics from the pet shop. These typically have detailed instructions included. Dose accordingly.