Here, the Torah uses the word “kallah.” Colloquially, we use that word to describe a bride but it literally means a daughter-in-law. Our verse says, “Do not uncover the nakedness of your kallah because she is your son’s wife.” So, the Torah is actually defining the word for us. (It wouldn’t make any sense if it actually meant that a groom couldn’t be intimate with his own bride!)
While she is not a blood relative, engaging in sexual relations with a daughter-in-law encroaches on one’s relationship with one’s son. As with other incestuous relationships, it makes no difference whether the son is born in or out of wedlock or from a forbidden union. Likewise, the law remains in effect even if the son dies or divorces his wife.
The rabbis instituted a preventive “fence” around this mitzvah by enacting a secondary ban on relations with a grandson’s wife and a great-grandson’s wife, etc. They likewise banned relations with a daughter’s daughter-in-law, though just for the one generation, as that less resembles the Torah’s prohibition. (Of course, any man’s wife is prohibited as adultery while they are married, so this must include the restrictions on post-death or divorce.)
A stepson’s wife is permitted (once the marriage is dissolved by death or divorce), as is the wife of a stepdaughter’s son.
This mitzvah applies in all times and places. It is discussed in the Talmud in tractates Sanhedrin (54a), Yevamos (21a-b) and Kerisos (2a). It is codified in the Shulchan Aruch in Even Ha’Ezer 15. This prohibition is #343 of the 365 negative mitzvos in the Rambam’s Sefer HaMitzvos and #115 of the 194 negative mitzvos that can be fulfilled today as listed in the Chofetz Chaim’s Sefer HaMitzvos HaKatzar.