Suppose you keep a goldfish in your house. One day, it dies. Assuming that it has died a natural death (that is, you didn't intentionally kill it), is it kosher to eat this goldfish?
Kosher mammals (cows, sheep, goats, deer, etc.) and birds (pigeons, chickens, etc.) must be killed by kosher slaughter. So if your pet pigeon dies of natural causes, it’s not kosher. (In fact, if you kosher-slaughtered it but then found it had a massive tumor that would have caused it to die soon anyhow of natural causes, it’s still not kosher!)
Kosher fish — including goldfish — require no special form of killing. The source listed in DoubleAA’s answer was “Rambam shechita” — “Rambam” refers to Rabbi Moses Maimonides, who wrote a code of Jewish law around the year 1200. One of the sections in that code is “Hilchot shechita”, laws of kosher slaughter. So the reference given was: See Maimonides’ code, laws of kosher slaughter, ch. such-and-such, paragraph such-and-such. (The sacred texts from which things originate are generally the Bible and Talmud; however, Maimonides’ code does a great job summarizing and organizing that. Hence you’ll see it quoted frequently here.)
The source that kosher fish do not to be killed specially is based on the Bible. An exasperated Moses says to God, “could sheep and cattle be slaughtered for them and that would be enough, or could all the fish of the sea be gathered for them, would that be enough?” Moses’ language makes clear that slaughter is unnecessary for fish, simply gathering.
Yes. Goldfish are kosher. You can even kill it to eat it if you want. Ritual slaughter is not necessary for fish (Rambam (Maimonides), Laws of Shechita (Slaughter) 1:3).