The Infinite Guitar was created by Michael Brook, as a way of allowing an electric guitar note to be held with infinite sustain (hence the name). It consists of an electronic circuit that takes the signal from a standard guitar pickup, amplifies it, and feeds it back into a separate pickup coil. When set up and used correctly, the result is a continuous sustained note that can be used as is, or treated to create new sounds or emulate traditional instruments.
In addition to his own instrument, based on a Tokai Strat copy, Brook produced two Infinite Guitars, one of which belongs to Daniel Lanois. The other belongs to The Edge of U2, who famously used it on "With or Without You" from 1987's The Joshua Tree.
The principle has been the subject of patent litigation in the past, and is currently available commercially in several forms, including the Sustainiac, the Fernandes Sustainer and the Vibesware Guitar Resonator.