We are not allowed to wear a garment containing both wool and linen. This forbidden mixture has a special name: shaatnez.
The reason for this mitzvah? Let’s start with a story. In Medieval times, there was a legend of a creature called the vegetable lamb. This myth sprung up as a way to explain the existence of cotton, which was not indigenous to the Old World. When travelers described wool that grew on a bush, the folks back home envisioned sheep that grew from a plant. To our sensibilities, isn’t such a hybrid creature nonsensical?
Well, that’s what shaatnez is. Wool comes from an animal and linen comes from a plant. Mixing them together is tantamount to second-guessing God by suggesting that His world is missing something that He “should have” created. This is also why we don’t crossbreed animals or graft different species of trees together. Wearing shaatnez is as nonsensical in the spiritual sense as the vegetable lamb is in the zoological sense.
The Rambam attributes the ban on shaatnez to an idolatrous practice that was still common in his time.
A person violates the prohibition of shaatnez every time he puts his head through the neck hole. If he is informed that a garment contains shaatnez and that he should remove it, he is in violation for each interval of time that would be sufficient to remove the garment in question.
This mitzvah applies to both men and women in all times and places. It is discussed in the Mishnah in tractate Kilayim; in the Talmud in tractate Makkos (21a) and elsewhere. It is codified in the Shulchan Aruch in Yoreh Deah 298. This mitzvah is #42 of the 365 negative mitzvos in the Rambam’s Sefer HaMitzvos and #181 of the 194 negative mitzvos that can be observed today as listed in the Sefer HaMitzvos HaKatzar of the Chofetz Chaim.