Fantails are a type of fancy goldfish. While there are differences between fantails and regular or "feeder" goldfish, both are different types or breeds of one species, Carassius auratus auratus. Selective breeding has given fantails many gorgeous traits not found in the wild type.
Plain and Fancy
Goldfish can be divided into two broad categories: plain and fancy. Plain goldfish are goldfish that more or less resemble their wild goldfish ancestors. But they are often available in several colors, including silvers and the famous golds. Fancy goldfish are goldfish that have been bred to be more ornate than their plain cousins. Fantails are a type of fancy goldfish, bred to have specific fin shapes.
What Makes a Fantail?
In the strictest sense, a fantail is any goldfish with a doubled caudal or tail fin. Since each caudal fin has two lobes, it gives the impression of four flowing tails coming off the fish. However, when you say "fantail" it's usually understood to mean a fish that has a fantail but otherwise looks like a plain goldfish. they are probably very close to the first fancy goldfish ever bred, having a mostly plain body and a fancy tail.
Other Breeds That Have Fantails
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The fantail was probably the starting point for fancy goldfish, so many other types of fancy goldfish also feature a fantail. You can find fantails in moors (black goldfish with telescope eyes), the humpbacked Ryukin goldfish and odd breeds like the bubble-eye goldfish. Though you wouldn't call them "fantails," these fish have the same tail shape.
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Plain goldfish are not that different from fantails with the exception of their smaller, less ornate tail fins. There are a few other differences. Plain goldfish are much less selectively bred. Because of this, their coloration can be a bit of a wildcard, while fantails tend to be more standardized. Plain goldfish tend to be much more robust than most fancy varieties, since they are less inbred.