The Tora says: "These you may eat, of all that are in the waters. Everything in the waters that has fins and scales ... you may eat. But anything in the seas or the rivers that has not fins and scales ... is forbidden to you" (Lev. 11:9-11). Both elements, scales (qasqeset) and fins (senapir) are necessary for a fish to be considered Kosher. The Rabbis of the Talmud taught that although both elements are necessary, all fish with scales also have fins. Thus, in practical terms, a Kosher fish is identified simply by the presence of scales (the scales are obviously removed in the process of cleaning the fish). Illustrations: Salmon, Tuna, White fish, etc. are Kosher (more specifically tabor) because they have scales. Crustaceans such as lobster or squid, clams, oysters, etc. are not Kosher (they are ta-me) because they lack scales.
SCALES: Some "scales" not considered Kosher by Jewish Law. For example, when the scales are part of the fish-skin and cannot be removed without damaging the skin. Illustrations: Sturgeon, although it has primitive bony-plates on its sides, is not Kosher because its scales (Gandoid scales) cannot be removed without damaging the flesh. Sharks are also non-Kosher fish. Their skin is covered with tiny teeth-like armor (placoid scales) which look like scales.
NAMES: When looking for a Kosher fish bear in mind that the names might vary from place to place. A given specie of fish might be known by five or more names and some of those names might sound Kosher. Illustration: We know that salmon is Kosher. But what about "Rock Salmon"? Is Rock salmon a standard variety of salmon, like Pink Salmon, Atlantic Salmon, Wild Salmon? Rock Salmon (very popular in England) is a non-Kosher fish, a small shark known as spiny-dogfish, which bears no relationship to the common Kosher specie of true salmon.
For a comprehensive list of Kosher and non-Kosher fish see this