There are two words for fat in Hebrew. Shumen refers to fats that one is permitted to eat, while cheilev refers to particular fats that are prohibited, even if they come from perfectly kosher animals. Cheilev is the fat that in the case of sacrifices was burned on the altar. It is prohibited to eat these fats even from non-sacrificial animals. (Basically, these are the fats on the kidneys and the intestines, etc. The fat tail of a ram is not considered cheilev and is permitted.)
The Sefer HaChinuch gives an explanation of this mitzvah similar to his explanation of the prohibition against eating mortally-wounded animals (Mitzvah #73): just as certain foods are physically harmful and a doctor would warn us about their dangers, other foods are spiritually harmful and God Himself cautions us not to eat them.
The Talmud in Kerisos (4a) explicitly says that only cheilev from these three species, which could be used as sacrifices, is prohibited. Analogous fats from other kosher animals, such as deer or giraffes, were not prohibited under this law. Additionally, if an animal is slaughtered and there is a fetus in it, the fetus may be eaten in its entirety, including the normally-prohibited fats (see Talmud Chulin 74a).
Eating cheilev is a serious offense, carrying a penalty of kareis (spiritual excision). These fats are removed from kosher meats in a process called nikkur (commonly translated as de-veining, as the sciatic nerve is also removed in the process; “treibering” in Yiddish) prior to the meat being offered for sale, so meat at your local kosher butcher is already cheilev-free.
This prohibition applies to both men and women in all times and places. It is discussed in the Talmud in the seventh chapter of tractate Chulin and in the first chapter of tractate Kerisos. It is codified in the Shulchan Aruch in Yoreh Deah 64. This mitzvah is #185 of the 365 negative mitzvos in the Rambam’s Sefer HaMitzvos and #88 of the 194 positive mitzvos that can be fulfilled today as listed in the Chofetz Chaim’s Sefer HaMitzvos HaKatzar.