Wikipedia
Fin rot is a symptom of disease or the actual disease in fish. This is a disease which is most often observed in aquaria and aquaculture, but can also occur in natural populations.
Fin rot can be the result of a bacterial infection ( Pseudomonas fluorescens, which causes a ragged rotting of the fin), or as a fungal infection (which rots the fin more evenly and is more likely to produce a white 'edge'). Sometimes, both types of infection are seen together. Infection is commonly brought on by bad water conditions, injury, poor diet, or as a secondary infection in a fish which is already stressed by other disease.
Fin rot starts at the edge of the fins, and destroys more and more tissue until it reaches the fin base. If it does reach the fin base, the fish will never be able to regenerate the lost tissue. At this point, the disease may attack the fish's body directly.
Fin rot is common in bettas due to poor water conditions in pet stores.