An instantaneous
teleportation device referred to in
Hyperion,
Fall of Hyperion,
Endymion, and
Rise of Endymion.
WARNING: Spoilers below!
Invented and implemented by the TechnoCore, the technology that drives the farcaster is so complex that only the AI's can understand the physics behind it. When a human travels via farcaster, they are suspended for a few milliseconds, a small enough time span that no one will notice. In this fraction of an instant, the neurons of the traveller are preyed upon by the TechnoCore in order to make calculations.
The "fall of the farcasters" occurs in Fall of Hyperion. Meina Gladstone orders FORCE:space to destroy all farcaster hubs that are in orbit above Hegemony of Man planets. This is an effort to slow down the Ouster onset.
Martin Selinus, among others, uses the farcasters to create a house that is on several different worlds at once. Only the grotesquely rich can afford this.
Later on in the series, Aenea activates the ancient farcaster to escape the Catholic Church.