Mammals, birds, fish, and even some locusts are kosher only if they meet the exacting criteria set out in the Bible and rabbinic law.
Reprinted with permission from The Jewish Religion: A Companion, published by Oxford University Press.
From two biblical passages (Leviticus 11 and Deuteronomy 14:3-21) the following rules are extracted regarding which animals, birds, and fishes are kosher and which terefah [non-kosher].
Only animals that have cloven hooves and that chew the cud are permitted. The pig does have cloven hoofs but does not chew the cud and is, consequently, forbidden. In the course of time, Jews came to have an aversion to the pig in particular, especially after Jews, in the period of the Maccabees [second century BCE] were ready to give their lives rather than eat pig-meat when ordered by tyrants to do so as an expression of disloyalty to the Jewish religion as a whole. Many a Jew today, otherwise not too observant of the dietary laws, will still refuse steadfastly to eat swine-flesh. It might be remarked, however, it is only eating of the pig that is forbidden. Surprising though this may seem at first glance, there is no objection, in Jewish law, to a Jew having a pigskin wallet.
The passage in Deuteronomy (14:4-5) gives a list of the animals that chew the cud and have cloven hooves and are thus kosher: oxen, sheep, goats, deer, gazelles, roebuck, wild goats, ibex, antelopes, and mountain sheep. It is interesting to note that whale meat and whale oil are forbidden not because the whale is a forbidden fish but because the whale is a mammal that, obviously, does not have cloven hooves and does not chew the cud.
With regard to birds, the Bible gives a list of the forbidden birds, implying that all others are kosher. But since the exact identity of the birds mentioned is uncertain, it is the practice only to eat birds that are known by tradition to be kosher, such as chickens, turkeys, ducks, geese, and pigeons. The eggs of forbidden birds are terefah, but quails' eggs are permitted since the quail is a kosher bird (see Numbers 12:31-2).
Nowhere in the whole of the Bible is there any reference to a particular fish, only to fish in general. In the two passages dealing with the dietary laws it is stated that only fish that have fins and scales are kosher. The Talmud lays down the rule that a fish that has scales also has fins, so that what actually determines which fishes are kosher is the existence of scales. A problem arises as to how "scales" are defined. [The medieval thinker] Nahmanides understands that only scales that are detachable from the skin of the fish qualify as scales. Where they cannot be detached they are not considered to be scales at all but part of the fish itself.
This is the reasoning behind a fierce debate that took place between rabbinic authorities in the eighteenth century regarding the permissibility of caviar, derived from the sturgeon, since the scales of the sturgeon cannot easily be detached from the skin of the fish, although they can be removed by the application of a lye solution. Some Orthodox Jews today do consider [such] caviar to be kosher, others do not.