A fascinating atmospheric strategy game involving the Earth being invaded by several alien races and a team set up to combat, capture, learn, and reverse engineer new weapons as the war expands. Introductory video here. One of the classics of PC games.
It looks like it is now an abandonware game available here and an Windows XP patch here. You will also need a processor slowdown tool like turbo or dosbox.
An encyclopedic overview and strategies for the game is here. An X-Com utility for the game is here. There is still a forum for the game here.
A closely related sequel X-Com - Terror from the Deep concentrates on the underwater alien bases. It is even more atmospheric and plays off of the H. P. Lovecraft Cthulhu Mythos.
X-Com Apocalypse was the much less popular final chapter in the series. It is interesting in that it draws a bit of inspiration from the Illuminati Mythos. This war is a later separate alien attack from another dimension that must be defeated. DOS - free download. Conversions to get it to run now.
The creators also produced Laser Squad Nemesis, more of a turn based strategy game playable by email. An old game and cracks are available if you go that route but it is less than $10 as shareware.