The Armageddon Clock is the name of the down counter of nuclear bomb drops remaining until the end of the world. That counter appears in the player(s) screen after the first nuclear bomb is detonated.
Armageddon is the term used to refer to the end of the known world, caused by Nuclear Warfare. In the game, it occurs when a certain number of Nuclear Missiles (determined by the number of players in the game) is launched. When this happens, the game ends, and all players lose the game with a score of 0. However, in some game modes, Armageddon does not appear, so you are free to launch as many nukes as you like.
If the usage of nuclear missiles is allowed, CPU players will continuously use these destructive weapons until all players are about to reach Armageddon.