Quite simply, it's having sexual activity with someone, anyone when you just KNOW Armegeddon is arriving, and I mean really FAST.
1. In the comedy movie "Spies Like Us", 2 stumblebum agents played by Chevy Chase and Dan Ackroyd along with a female agent also hired by the U.S. military (Dan's real-life wife Donna Dixon) tangle with a Soviet Army team at a Soviet ICBM launch team. In the process the nuclear missile gets launched by mistake. Everyone there whether they be from the U.S.A. or the U.S.S.R thinks that this error will spark the nuclear suicide of planet Earth so everyone decides to have apocalypse sex when the end comes. Dan motions for Donna to enter his tent nearby, she smiles and joins him. Chevy Chase gestures for a female Russkie beauty to be his death partner, she joins him. The remaining 2 in the group are both Russian men - one grins like a child. Yeah I know - a dopey old homophobic joke about foreigners - hyuk hyuk. The ICBM malfunctions in the sky anyway.
2. The Tears For Fears song "Famous Last Words" is about a romantic couple being together when nuclear doomsday is imminant. The song doesn't say if the two are having apocalypse sex, however.
3. Apocalypse sex is what happens when two lovers hold on each other tight at the end of the world. On the other hand, on an episode of M*A*S*H*, a TV show set during the Korean War, Dr. Benjamin Franklin "Hawkeye" Pierce and Major "Hot Lips" Hoolahan, who normally can't stand each other, are trapped in a shelter during intense bombing and they have a sort of "apocalypse sex" because they fear they won't survive the night's shelling.