The ALEX 7000 computer was developed by Elijah Cooper and principally put into use at the Dakota Complex. He had his own operating system and was equipped with an artificial intelligence. "Alex" maintained control of the entire facility, guiding all other hardware and computers to do Elijah Copper's bidding.
Alex was programmed to successfully triumph over any intruder attempting to deactivate the nuclear weapon inside of the complex. Jaime near the end of the episode (Part II) informed Alex that there was no such weapon. However Alex processed this as a sign that he had not triumphed over Jaime; he could not quantify this or truly understand it and planned to use the B-52 (nuclear weapon) that Oscar was sending to end life on this planet as Elijah "intended". Jaime is forced to destroy ALEX by activating a fire sprinkler system, causing the AI to short out.
Some of ALEX's components were later salvaged by the OSI and placed in its vault.
The glowing eye inside of the black sphere (as well as ALEX's name) is a homage to HAL 9000; an artificial intelligence that starred in Stanley Kubrick's "2001: A Space Odyssey" which was a depiction of the future with a surreal approach. Kenneth Johnson acknowledges this in the DVD commentary.