Here is a brief description of the types of pond goldfish:
Common or hibuna: Hibuna is the Japanese term for what we call the common goldfish. (In Japan, the wakin is the "common goldfish.") The hibuna is a colorful version of the wild carp ancestors of the goldfish. The tail is short (not more than 1/4 the body length) with rounded lobes. It comes in all shades between true red and lemon yellow, with orange the most common. It can be red/orange/yellow with black, although the black color is often unstable. It can be solid white or mixtures of white and red/orange/yellow. A calico hibuna is called a London shubunkin.
Comet: Comets originated in the US. They are similar to commons, but have a somewhat more slender body than the hibuna and have a long forked tail. Ideally the tail should be as long or longer than the body in a mature specimen, but if the tail is too long to be a proper hibuna and has pointed lobes, it's called a comet. They come in the same colors as the hibuna. Red and white comets are called sarasa comets. Calico comets are called shubunkins, American shubunkins, or Japanese/American shubunkins.
Shubunkins: In addition to the two types of shubunkins listed above, the Bristol shubunkin has the body form of the hibuna with a large heart-shaped tail.
Wakin: This is the "common goldfish" of Japan. The body shape is like the hibuna with a slight downward tuck at the tail. The tail is double, rather than single like the varieties above. It comes in all the colors of the above varieties. The red in red and white wakins is often particularly vibrant. It also comes in calico, and that is called a calico wakin.
Watonai: Watonai are basically long-tailed wakins. They come in all the colors of the above fish.
Jikin: The jikin is like the hibuna in body shape, but has a fully split double tail. From the back, the tail looks like a four-leafed clover. It's not common in the US, but is considered a pond fish in the far east.
You can see all of these as well as many kinds of fancy goldies at the
Rain Garden site.