What effect did compression waves have immediately following the Big Bang?
In the very early Universe there were regions of high and low density that arouse from a period of inflation in the first fraction of a second. These density fluctuations generated compression waves, very much analogous to normal sound waves, but travelling through the hot plasma of the early Universe. The fundamental thing forcing the waves was the high energy photons (light) emanating from the regions of high density. Light is scattered and blocked by hot plasma, so its energy and momentum can push the wave onwards.
However, when the Universe was around 400,000 years old it became cool enough for electron and protons to stably bind together to form hydrogen (rather than plasma). This meant that at this point the driving force behind the waves stopped pushing, so they ceased to propagate. Today we see the distance they had propagated as about 500 million light-years (after accounting for the expansion of the Universe between then and now).
As a result of all this, when astronomers perform galaxy surveys today you are slightly more likely to find pairs of galaxies separated by around 500 million light-years than you might otherwise expect. This phenomenon is known as baryon acoustic oscillations.