Fish usually do things for a reason, though sometimes that reason stems from fish-logic. Inter-species aggression -- which can take the form of chasing other members of your species around -- grows from real-world concerns. While it can be inconvenient in your fish tank, aggression is useful in the wild
Basis of Aggression
In terms of survival, aggression carries a trade-off. It costs energy to chase and fight other animals, including members of your own species. However, in some environments, the benefits of aggression outweigh the downsides. Biomes like coral reefs have intense competition for food and space. Aggression gives angelfish the edge they need to carve out a space of their own to graze for food and breed.
Unlike their saltwater counterparts, freshwater angelfish rarely display aggression. However, when breeding, they suddenly start chasing away other fish, including other angelfish. In this case, the aggression stems from defending their young. This behavior ensures that angelfish will be able to pass on their genes. Freshwater angelfish belong to the cichlid family, a group of fish that almost uniformly displays an unusual amount of parental care for a fish. Freshwater angels will also "fan" fresh water onto their eggs and pick out dead or unfertilized eggs in between chasing other fish away.
Saltwater angelfish tend to bring a lot of aggression to bear. Across the different species of angelfish -- and even individuals within a species -- their level of aggression varies somewhat. But in general, adult males tend to be more aggressive than females and juveniles. Most often, males will chase away other males of the same or similar species. Marine angelfish live and breed in harems of one male and many females. So chasing away other males ensures that the dominant male gets to pass his genes on.
Some things can short-circuit aggression in male angelfish. Often, aggressive males will chase and fight males of similar shaped or colored angelfish. Sometimes, they will even go after non-angelfish with similar patterns. For this reason, you have to take care when selecting tank mates for your angelfish. In most species, you cannot keep more than a single male, and maybe a female, to avoid aggression.