Meat from kosher animals is permitted. Milk from kosher animals is permitted. Meat plus milk, even if both are from kosher animals? No go.
Meat and milk combinations are prohibited several times in the Torah. Here, the Torah specifically prohibits the act of cooking them together. The terms, however, are not meant to be specific. “A kid” does not limit this mitzvah to young goats. As the Talmud explains In Chulin 113a-b, when the Torah says “g’di” (a kid), it means any type of animal. When it wants to refer specifically to a goat, the term used is “g’di ha’izim” (“a kid of goats,” as in Genesis 38:20).
There are a number of ideas underlying this mitzvah, but none of them are based on health. (Yes, it affects one’s spiritual well-being, but that’s true of all mitzvos.) The Sefer HaChinuch compares this mitzvah to those forbidding grafting, crossbreeding and the like. In those mitzvos, the act is an affront to God as the person attempts to bring something new into the world that He did not create. Similarly here, God created milk and He created meat and never the two shall meet. To combine them is tantamount to crossbreeding and other acts that imply that somehow something is lacking in God’s work. (The Sefer HaChinuch supports this position by pointing out that it is prohibited simply to cook the mixture, even if one doesn’t eat it. We’ll discuss the prohibition against eating such a combination in Mitzvah #113.)
This mitzvah does not only apply to the milk of an animal’s own mother. In this author’s view, the mixing of dead animal flesh with milk, which is used by a mother to nurture her young, is grossly insensitive. By using the stark and compelling imagery of cooking a kid in its own mother’s milk, the Torah drives home just how insensitive that would be.
The prohibition against cooking meat and milk together is in effect in all times and places for both men and for women. It is discussed in the eighth chapter of the Talmudic tractate of Chulin and codified in the Shulchan Aruch in Yoreh Deah 87. It is #186 of the 365 negative mitzvos in the Rambam’s Sefer HaMitzvos and #91 of the 194 negative mitzvos that can be fulfilled today as listed in the Chofetz Chaim’s Sefer HaMitzvos HaKatzar.