Flanging is a time delay effect in which the dry signal is mixed with a briefly delayed version of itself. 
If this short delay is in the region of 0 - 20 milliseconds the ear will not be able to resolve the two signals separately but will hear them as one signal with phase cancellations. 
The flanging effect is made most noticable when the delay time is modulated by a low frequency oscillator (LFO). This has the effect of causing the frequencies of the phase cancellations and accentuations to shift up and down the frequency spectrum providing the characteristic "swishing" effect. 

Flanging is aurally very similar to phasing which creates a similar effect through the use of phase-shift circuitry. 

an audio treatment where the input signal is delayed slightly (.1 - 20 milliseconds) and mixed with the original. this creates complex phase cancellations and reinforcements which results in dynamic changes to the signal. to accentuate this, the pitch of the delayed signal is usually modulated by a sub audio sine or triangle wave.  the output is fed back into the input to create a small amount of whine. the amount of modulation is called width and the depth refers to the amount of feedback.  the term flanging refers to the pre digital technique of using two tape decks for this effect: finger pressure on the flange of one of the tape reels would cause the necessary speed variations.
The flange effect can be altered by inverting the polarity of the delayed signal with the result that the lower frequencies will be virtually cancelled out leaving the effect in the higher frequencies only.  A portion of the effected signal can be fed back into the delay to reinforce the frequency colouration by increasing the dips and peaks. 
Sound Examples:
  clarinet    clarinet (flanged) 
See Also:
Chorusing   Doubling   Digital Delay   echo