A moving sound source comes from a source that is moving in any direction from the point where sound is being recorded. Because of the doppler effect, the pitch and volume of the moving sound source changes. The Doppler effect is when the object is coming towards you, sound waves infront of the moving object become compressed, causing a smaller wavelength which causes a higher frequency. When the object is going away from you the sound waves become more spread out, creating a long wave length and a lower frequency. the easiest way to see this is to watch a passing vehicle, especially at high speeds.
Here is a perfect example of the Doppler effect and how the sound of a moving object varies.
Sometimes if the recording is directed through stereo headphones or speakers the levels of sound played from right to left or left to right (depending on the direction of the moving object) will vary, so that it seems like the object is actually moving past the viewer.
It is necessary to remark that the definition of stationary source you are given is not exactly correct, and It could be confusing. You are talking about stationarity in position, but he term "stationary sound source" refers to the statistical characterization of the source. So, in a "wide sense" a stationary sound soruce is that which has the same mean and variance over time.
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