In most situations, a damaged tail fin will readily regrow on an aquarium fish. However, you need to figure out what damaged the fin the first place. Identifying the cause will enable you to help prevent recurrence and encourage healing.
Some fish love to fight, to the detriment of the fins of both the aggressor and the victim. For example, cichlids and bettas are notorious for attacking other fish, damaging their fins, tails and other parts. In these species, aggression is related to territorial behavior. However, some fish express aggression unrelated to territory. For example, barbs, schooling fish, will nip at the fins of other species, particularly when barbs are in groups of fewer than six. As long as an injured fish is not under constant attack, the fish will usually heal and regrow a damaged tail fin quickly.
Even without boisterous tank mates, fish can injure their fins in a number of ways. Fish may get injuries to their tail fins when netted for transport. Fish may also injure their fins by getting them caught on the decorations or substrate in fish tanks. Aquarium fish can even hurt their fins by bumping into the aquarium glass. Like injuries from fights, fish with these types of injuries tend to heal quickly and regrow their tails without incident.
Normally, injured fins have a clean look to them. Ripped fins tend to have clean splits, and nipped fins look like bite marks. However, injuries to the fins can leave fish open to opportunistic infections. Fin rot occurs when a fungus or bacteria infects an injured fin. Fin rot typically has a ragged fuzzy look to it. This type of infection may also cause slimy patches or red streaks of the fin. Fin rot needs rapid action on your part to keep it from getting worse. Fin rot can rapidly destroy a fish's fins. A fish infected with fin rot will not regrow its tail unless the infection is cured and the fish is provided with clean aquarium water.
If you do not see signs of infection, you do not have to treat damaged tail fins. In these cases, fish will rapidly heal and regrow injured fins. However, you may need to take steps to prevent recurrence, like moving aggressive fish to another tank to give the injured fish time to heal. Regular water changes can improve the health of fish and encourage healing and prevent fin rot. If you see fin rot, procure some aquarium antibiotics and treat them per the manufacturer's instructions. If you take these steps, fish can usually regrow their injured tail fins.