Dynamic Time Warping is pretty well explained on this site. I'll use some of the diagrams from the PPT on that site to explain.
The idea is to divide the signals into segments (frames) and then compare frames sequentially through each signal. As illustrated below, motion from a segment in one signal to the next segment depends on the similarity to the current segment in the other signal.
For your example of audio files, you usually have features that you extract from them (LPCs or MFCCs for example). Extract these from each frame. Use these features as the comparison features used in the DTW algorithm.