ReplayGain is the name of a technique invented to achieve the same perceived playback loudness of audio files. It defines an algorithm to measure the perceived loudness of audio data.
ReplayGain allows the loudness of each song within a collection of songs to be consistent. This is called 'Track Gain' (or 'Radio Gain' in earlier parlance). It also allows the loudness of a specific sub-collection (an "album") to be consistent with the rest of the collection, while allowing the dynamics from song to song on the album to remain intact. This is called 'Album Gain' (or 'Audiophile Gain' in earlier parlance). This is especially important when listening to classical music albums, because quiet tracks need to remain a certain degree quieter than the louder ones.
ReplayGain is different from peak normalization. Peak normalization merely ensures that the peak amplitude reaches a certain level. This does not ensure equal loudness. The ReplayGain technique measures the effective power of the waveform (i.e. the RMS power after applying an "equal loudness contour"), and then adjusts the amplitude of the waveform accordingly. The result is that Replay Gained waveforms are usually more uniformly amplified than peak-normalized waveforms.