Under normal circumstances, a goldfish tail will grow back, depending on how much of the tail has been lost. However, regrowth is also dependent upon an appropriate environment.
If part of a tail is broken or bitten off, it will appear torn or split. Often, this type of injury is the result of an attack by another fish. If the break has a ragged, fuzzy edge to it, the loss might be the result of a bacterial infection. In either case, the most important way to promote healing is to provide clean water. Clean water is healthier for the fish and will make it harder for a bacterial infection to spread.