Ascended Meme: Many fans of the original X-Com loved to exaggerate the threat of Chryssalids, several Let's Play retellings of playthrough deliberately injecting missions where X-Com is forced to face massive numbers of Chryssalids, most notably Guava's LP where Japan is overrun. The Enemy Within Council Mission "Site Recon" plays out almost exactly like some of them, even allowing for the possibility of operatives failing to reach the evac point by the time bombs start dropping!
Code Black: Refers to a failed mission where all your operatives got horribly murdered by aliens or EXALT.
C-Hive, C-Day: C-Day refers to the first time you encounter Chryssalids, an old term inherited from the original X-Com fandom. C-Hive refers to the Chryssalid 'hive' aboard the fishing vessel in the Council Mission Site Recon. Sometimes Site Recon itself can be addressed as C-Day on its own.
Thin Mints: Thin Men deliberately misspelled.
Crit Men: An infamous nickname for Thin Men from higher difficulties, where players find them distressingly accurate.
Running the Asylum: Pretty much everyone on the development team adored the original. Enforced for those who hadn't played it before; every other Friday, time was set aside to play the original. All of them became addicted.
A story Quentin Smith of the "Sit Down and Play" site told was that in the early stages of development, the developers couldn't figure out how the new XCOM was supposed to play. So they went to the house of Sid Meier and used dice, paper and wooden tokens as a bare-bones XCOM to get a feel of it. An XCOM board game... Wouldn't you wanna get your friends around a table to give it a shot?
Originally, the Muton was supposed to get a mechanical variant in Enemy Within instead of the Sectoid, but was looked over due to already being big.
The base ini file suggests there were as many as 36 nations on the Council before it was pared down to 16. The remnants of this can be seen in the appearance of nationalities among recruited soldiers not on the Council such as the Netherlands, Sweden, and South Korea.