These are the species people ask about most, and they all fail the same test in different ways. Here is the why for each.
Sharks
Sharks have no true scales. What looks like skin is covered in tiny tooth-like structures, not removable scales. No scales means not kosher, even though sharks have fins.
Catfish
Catfish are smooth-skinned with no scales at all. They have fins, but with zero scales they fail the test. Catfish is one of the most common non-kosher fish sold at regular markets, so it is worth knowing on sight.
Eels
Eels look snakelike and have no scales you can remove in the kosher sense. They are not kosher. This includes both freshwater eels and the saltwater species sold for unagi in sushi.
Shellfish and other sea creatures
Shrimp, lobster, crab, clams, oysters, mussels, scallops, octopus, and squid are not fish in the kosher sense and have no scales. None of them are kosher. The fins-and-scales rule is about fish, and these fall outside the category entirely.
The pattern
In nearly every case, the missing piece is scales. If you remember to check for removable scales, you can predict most of these answers yourself. The site confirms it for you on each fish page.