For sometime it had been known that the Asgard were a dying race, and after acquiring a genetically deteriorating disease from their last attempt to solve their problem, the Asgard accepted that there was nothing left they could do. The computer core was their legacy, and a way to ensure the Asgard were forever remembered by the intergalactic community.
With the extinction of the Asgard race imminent, the Asgard installed a sophisticated computer core on the Tau'ri ship Odyssey. This computer core contained all of the Asgard's most up-to-date knowledge as well as their entire recorded history. Included in the computer core was extensive information on Asgard beaming, time dilation, shielding, weapons, hyperdrives and much more, and powered by an Asgard power core; with a time dilation device integrated into the core itself. The core is very user friendly, and a user could download a simple program instructing the core to perform a very broad function without requiring a detailed knowledge of the specified function; the core would "fill in the blanks". The computer core could also present a holographic image of any Asgard on record as an interface, intended to assist any who do not understand the Asgard language and/or the technologies themselves. (SG1: "Unending", "The Ark of Truth")
By the year 2008, an Asgard core had been installed on the Daedalus, and likely on other Daedalus-Class warships. This version was however aesthetically different, so it is possible that they differ from the original in some way (ATL: "Search and Rescue").