This seems pretty simple. If you have a sheep, don’t try to breed it with your goat. If you have a lion, don’t try to mate it with a tiger. Reindeer and gazelles? Wolves and foxes? Pandas and koalas? You can put those ideas right out of your head.
While all of the above examples above happen to include mammals, this mitzvah applies equally to crossbreeding birds, sea creatures, etc. So don’t try to mate a fruit bat with an octopus or anything else you can imagine – they’re all equally prohibited.
It is permitted to mate two things of the same type even if they’re dissimilar. For example, different species of cattle are all still cattle. So breeding cows and horses together is out but a Guernsey with a Jersey cow is probably okay.
A mule is the offspring of a male donkey and a female horse. (The offspring of a male horse and a female donkey is called a hinny and is more uncommon.) Mules are generally infertile. Halachically, horses, donkeys, mules and hinnies are all considered different species, so don’t try to mate any of them with any others.
The reason underlying this mitzvah is simple enough: God created the world and everything in it. He put reindeer and gazelles here, but not reinzelles. He created pandas and koalas, but not pandalas. For us to try to breed new species is tantamount to us saying, “It’s a pretty good world, but you know what it’s missing…?” What chutzpah!
As we discussed in Mitzvah #210, the prohibition against bestiality, God told every species to reproduce only with its own kind (Genesis 1:11). And, as we also stated there, the animals prior to the flood mated outside their species, which is why they were considered “corrupted” (Sanhedrin 108b). (In the movie “Ghostbusters,” when describing “a disaster of Biblical proportions,” among the fire, brimstone, darkness and earthquakes, Bill Murray famously included the example of “dogs and cats living together.” Yes, it’s actually that big a deal.)
This mitzvah applies to both men and women in all times and places. It is discussed in the Mishna in the first and eighth chapters of tractate Kilayim; in the Talmud, see Baba Kama 55a. This mitzvah is codified in the Shulchan Aruch in Yoreh Deah 297. It is #217 of the 365 negative mitzvos in the Rambam’s Sefer HaMitzvos and #142 of the 194 negative mitzvos that can be fulfilled today as listed in the Sefer HaMitzvos HaKatzar of the Chofetz Chaim.