An animal that died on its own, without having been properly slaughtered, is called a neveilah and it may not be eaten. It may, however, be sold to non-Jews (assuming, of course, that they're inclined to purchase it).
From the fact that this mitzvah specifies that we may sell it to non-Jews - a fact that should be obvious, since we already know from Leviticus 7:24 that we may benefit from such animal carcasses - the Sages of the Talmud derive laws pertaining to non-kosher ingredients that impart a detrimental taste, such as the negative aftertaste that dish soap might leave in your chulent. (See Talmud Avodah Zara 65b for how.)
The reason for this mitzvah is the same as the prohibition against eating a mortally-wounded animal (Mitzvah #73): God knows what foods may not harm a human body but which may be detrimental to a Jewish soul.
This mitzvah applies in all times and places. It is discussed in the Talmud in tractate Chulin on pages 32a-b. It is codified in the Shulchan Aruch in Yoreh Deah 17. This mitzvah is #180 of the 365 negative mitzvos in the Rambam’s Sefer HaMitzvos and #86 of the 194 negative mitzvos that can be observed today as listed in the Sefer HaMitzvos HaKatzar of the Chofetz Chaim.