In comparison to meat, the kosher requirements for fish seem tame. The Torah's specifications for kosher fish are fins and scales (Leviticus 11:9-10). Fish needs no ritual slaughter or salting. But it still requires kosher supervision.
If you take your kosher fish home from the market while it's still whole, then little can go wrong. But most fish today is processed in some way before it gets to you. Most fish markets use the same knives to cut both kosher and non-kosher fish, which creates kashrut problems. It's best then to buy your fresh fish from a store selling only kosher fish.
One complicating factor in kosher supervision of fish is that the Torah requires the kosher signs of each fish to be ascertained before any Jew may eat it. So proper supervision requires the constant presence of a conscientious kashrut supervisor throughout the process to check each fish before it is cut up and processed.
Canned fish, like tuna, is also a problem. Some of it is canned on-board or it is brought for canning in lands like Thailand or the Philippines. It's not easy to put a Torah-observant Jew as a supervisor there. And then before a canning plant can be certified, that plant must be made kosher too.
Soon, however, we will get to eat the most illustrious fish of all, the fabled Leviathan, as the Talmud says, at the great national feast after the revelation of the Mashiach. But we can be sure that that fish will have no kashrut problems, for it will be under the strict supervision of the Almighty Himself!