If one of the animals that you may eat dies, the person who touches its carcass shall be impure… (Leviticus 11:39)
A neveilah is a kosher animal that died without shechita (ritual slaughter). Such an animal carcass is inherently unclean and capable of transmitting this impurity to those who touch it. An olive-sized piece of such a corpse is all that is necessary to contaminate a person or a vessel. As with the eight sheratzim (Mitzvah #159), a neveilah will contaminate an earthenware vessel through its interior airspace. Unlike the eight sheratzim, a neveilah will also render impure the clothes on one who carries it.
One who becomes impure in this fashion may not enter the Temple, nor may he eat sacred food. After nightfall, he had to immerse himself in a mikvah, plus his clothes if he carried the carcass. It wasn’t forbidden to become impure, even on purpose, one just had to follow the rules if he did so.
None of this applies to a kosher animal that has been ritually slaughtered; such an animal is ritually pure.
This mitzvah applies to men and women in all times and places only in that such animal carcasses impart impurity. Beyond that, there’s no real practical application in the absence of the Temple. The details of this mitzvah are discussed in the Mishna in the tractates of Keilim and Tohoros. In the Talmud, see the fourth chapter of tractate Chulin, starting on page 70b, and the ninth chapter, starting on page 117b. This mitzvah is codified in the Mishneh Torah in the first chapter of Hilchos Shar Avos HaTumah and is #96 of the 248 positive mitzvos in the Rambam’s Sefer HaMitzvos but is not in the Sefer HaMitzvos HaKatzar of the Chofetz Chaim.