The Asgard computer core was a computer core containing all of the collective knowledge and technology of the Asgard.
Realizing that their race was dying, the Asgard chose to gift all of their knowledge and technology to the Tau'ri of Earth as the inheritors of their legacy and the Fifth Race. The effort was spearheaded by Supreme Commander Thor and was agreed upon after a report by Asgard engineer Hermiod who served aboard the Daedalus and was assigned to Lt. Colonel John Sheppard's team for a mission to determine whether or not humanity should receive the Asgard's collective knowledge. (SGA: "Hermiod's Last Mission")
In 2007, after over a year of work, the Asgard called the Tau'ri battlecruiser Odyssey to Orilla to gift them with the computer core and upgrade the Odyssey itself with their latest shields and weapons. Shortly after the computer core was installed, three Ori warships arrived and the Asgard destroyed Orilla in an act of mass suicide. Though the Odyssey succeeded in destroying an Ori warship with the new Asgard weapons, the Ori warships were able to track the Odyssey through an energy signature given off by the computer core detectable from subspace. A battle was fought to protect the computer core as Lt. Colonel Samantha Carter struggled to separate the Odyssey's hyperdrive controls from the computer core so it could be shut down. Due to the events of an alternate timeline, Carter ultimately succeeded and the Odyssey escaped with the computer core. (SG1: "Unending")
During the Battle in the Alteran Home Galaxy, the computer core was reactivated by James Marrick of the IOA to create a Replicator with which to attack the Ori fleet. After the Replicator escapes and begins multiplying, Carter is forced to keep the computer core online in order to find the Replicator self-destruct command despite it drawing Ori warships to their position. Carter was able to use the computer core to control the Odyssey's systems until they were taken over by the Replicators. After Marrick gives up the location of the self-destruct command, Carter is able to activate it and destroy the Replicators. (SG1: "The Ark of Truth")
After encountering the Asgard Fenrir in the Pegasus Galaxy, Carter has the Odyssey crew search for information on him in the core. Failing that, she sends Doctor Rodney McKay and Lt. Colonel John Sheppard to search. McKay is eventually able to locate information on Fenrir, including holographic records of his trial before the Asgard High Council heavily encrypted in a part of the core dedicated to the Asgard correctional system. (SGA: "Nightfall")
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, matter conversion and much more. The core was 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 upload a simple program instructing the core to perform a very broad function without requiring 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")
The Asgard computer core also contained a detailed record of Asgard history which could be accessed to find information on the Asgard as a whole or on more specific subjects of Asgard history. At least part of this, the part on the Asgard version of a correctional system, was encrypted, possibly due to the Asgard feeling embarrassed about this part of their history. At least one part of this history is shown to have detailed holographic recordings attached to it from real-time sensor logs to a visual trial record. (SG1: "Unending") (SGA: "Nightfall")
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. (SGA: "Search and Rescue").