These you may eat of everything in the waters: that which has fins and scales… (Leviticus 11:9)
Just like kosher animals have the signs of cud-chewing and completely split hooves, kosher fish also have signs: they must have fins and removable scales.
Remember how some animals have only one kosher sign or the other? Well, fish are different. Sure enough, some fish have fins but not scales (for example, sharks. Yes, sharks are fish). However, no fish has the proper type of removable scales unless it also has fins. In fact the Talmud tells us (Niddah 51b) that if we see the scales, we know that the fish is kosher and may eat it. This, of course, then raises the question of why we must be told fins at all.
One explanation the Talmud gives in discussing this question is that the average person may not know what snapir and kaskeses are. If the Torah had only said kaskeses, one might have thought that meant fins, with the result that he avoids shellfish but erroneously eats catfish. Since the Torah says both kaskeses and snapir, everyone knows that one is fins and the other is scales. (For the record, snapir means fins and kaskeses means scales.)
There’s more to the Talmud’s discussion than that; see there if you care to see how it proceeds. (Aside: a certain Rav once told this author that fins and scales are more than the just the signs of a kosher fish, it is the combination of fins and scales that causes a fish to be kosher.)
As with other laws pertaining to kosher foods, the reason is to keep us far away from things that are spiritually harmful, as we have said.
This mitzvah applies to both men and women in all times and places. It is discussed in the Talmud in tractate Chulin (66a-67a), as well as in the aforementioned Niddah 51b. It is codified in the Shulchan Aruch in Yoreh Deah 81. It is #152 of the 248 positive mitzvos in the Rambam’s Sefer HaMitzvos. As with checking the signs of kosher animals (Mitzvah #153), the Ramban does not consider this a separate mitzvah, nor is it listed in the Sefer HaMitzvos HaKatzar of the Chofetz Chaim.