17th of Av, 5770
The Torah says that fish or any creature that lives in the water must have both scales (kaskekset) and fins (senapir) in order to be considered Kosher. The rabbis of the Talmud teach us that all fish that have scales also have fins, so in practice Kosher fish are identified simply by their scales. Obviously, crustaceans (such as lobster) and other shellfish (such as clams) are not Kosher because they lack scales.
Not all scales are considered Kosher by Jewish Law. It must be removable without damage to the skin of the fish. Sturgeon, although it has primitive bony plates on its sides, is not considered Kosher because the scales cannot be removed without damaging the flesh. Sharks are similarly not Kosher, because their skin is covered with tiny teeth-like armor, which are not considered scales at all.
Jewish Law requires only a minimum number of scales to accord a fish Kosher status. Tuna, for example, have very few scales, yet are nevertheless considered a Kosher fish.
There are a few more factors that complicate this determination. For example, a given species of fish may be known by five or more names, some of which are common to known Kosher species. "Rock Salmon", for example, is a non-Kosher fish (otherwise known as Atlantic Wolfish), and bears no relationship to the common Kosher species of true salmon.
For a list of Kosher and non kosher fish see: http://www.kashrut.com/articles/fish/
Adapted from The Fortunes of a Fish by Rabbi Blech.