An Infinity Circuit is a psychic network contained in the wraithbone skeleton of an Eldar craftworld. They transmit psychic energy throughout the ship to power on-board devices, and since the fall of the Eldar, they also serve as artificial afterlives for Eldar souls, since otherwise they would be devoured by Slaanesh. All Infinity Circuits are in turn linked through the Webway, forming what is known as the Eternity Matrix.
When Eldar die, their souls are trapped by the spirit stones that they wear. These spirit stones are then placed in contact with the core of their craftworld's wraithbone skeleton, which draws the soul out and into the network. The soul's energy is added to the wraithbone's own, while its "personality" finally is laid to rest, safe from Slaanesh forever (as long as the integrity of the circuit itself is maintained—that's what the craftworld's Guardians and other armed forces are for). The hope is that one day, there will be enough psychic energy contained in all of the craftworlds' Infinity Circuits that a new Eldar god will form in the Immaterium, one powerful enough to destroy Slaanesh and save the Eldar race.
Infinity Circuits can also be sources of soldiers. If the need is truly dire, Spiritseers can guide souls out of the Infinity Circuit and back into their spirit stones, which can then be placed into wraithbone constructs like Wraithguard, Wraithlords, Wraithseers, Wraithknights and even tanks to fight for the craftworld once again. Eldar don't like doing this, because they prefer to leave their ancestors in peace, but sometimes it just has to be done. Craftworld Iyanden probably uses the most such "ghost warriors" after sustaining terrible losses against Hive Fleet Kraken; they put themselves through so much angst over it that the Dark Eldar found it funny. Certain tyranids find them delicious.
Most souls do not have any sort of awareness in the Infinity Circuits, their personalities essentially fading into its background energy (although like most things, it seems to vary from Craftworld to Craftworld, or rather writer to writer). An exception is made for those souls who deaths brought them exceptional grief or rage, who carry on their emotions through. The souls of Eldar with exceptionally strong personalities, typically a great hero of some kind, still maintain their awareness, and can even be consulted by Spiritseers for advice. The most aware are those Farseers who reach the end of their lifespans and become Crystal Seers, who while linked to the Infinity Circuit essentially are otherwise fully mentally alert. These levels of awareness roughly correspond to the power of their corresponding wraith-constructs; your standard soul becoming a Wraithguard, raging restless souls become Wraithblades, while great heroes become Wraithlords.