Flanging is an audio effect that occurs when two identical signals are mixed together, but with one signal time-delayed by a small and gradually changing amount, usually smaller than 20 milliseconds. This produces a swept 'comb filter effect'1: peaks and notches are produced in the resultant frequency spectrum, related to each other in a linear harmonic series. Varying the time delay causes these to sweep up and down the frequency spectrum.
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The development of the classic "flanging" effect is generally attributed to Ken Townsend, an engineer at EMI's Abbey Road Studio, who devised the process in the spring of 1966. Tired of the laborious process of re-recording dual vocal tracks, John Lennon asked Townsend if there was some way for the Beatles to get the sound of double-tracked vocals without actually doing the work. After mulling the problem over, Townsend devised Artificial Double Tracking2
...whenever Lennon wanted a Beatles song double-tracked, he would ask for "Ken's flanger". According to Lewisohn, "The Beatles' influence was so vast that the term "flanging" is still in use today, more than 20 years on."
1 - In signal processing, a comb filter adds a delayed version of a signal to itself, causing constructive and destructive interference.
The frequency response of a comb filter consists of a series of regularly-spaced spikes, giving the appearance of a comb.
2 - Automatic double tracking was an analogue recording technique designed to enhance the sound of voices or instruments during the recording process. It used tape delay to create a delayed copy of an audio signal which was then combined with the original.