Flanging is a type of phase-shifting. It is an effect which mixes the original signal with a varying, slightly delayed version of the signal. The original and delayed signals are mixed more or less equally.
Flanging results in a sweeping sound — see the following example (MP3):
The term flanging comes from the days of reel-to-reel tape recording. The original signal was recorded on a second reel, and the delay was achieved by holding a finger or thumb on the edge (flange) of the reel to physically slow it down. Flanging was made popular during the psychedelic music era in the 1960s and 1970s.
The following example is the flanger settings window in Adobe Audition. It shows some of the settings commonly used in flanging: