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Modulation, in music, is the phenomenon of changing the tonality of the music--that is, changing what key it's in.
In western classical music tradition (on which much of today's popular music worldwide is based), modulations are usually between keys that are close to each other--that is, they differ by at most one or two sharps or flats, if at all. One prominent exception to this is going between the major and minor keys of the same home note (e.g. C major and C minor) - but that's not really a modulation anyway, just a mode shift.
Going into the 19th century, musicians started to experiment with more stuff (culminating in the 20th century where people sometimes didn't even have a key at all). The resulting repertoire of common modulations grew quickly.
Usually, you can tell a modulation if the "flavor" of the music (and usually, though not necessarily, the emotion) changes. This doesn't happen too often in modern popular songs, partly because they're rather short, but can sometimes be found between verses and refrains, with verses in one key and refrains in another (examples including The Beatles' "Good Day Sunshine" and Midnight Sons' "If Only Tears Could Bring You Back".
Modulations are VERY common in classical music, being a staple of tonal music (i.e. dating from the late 1500s forward, as opposed to modal music, which cannot modulate by virtue of not having a key in the first place). They occur both in miniature (lasting only a couple bars, or even a couple beats) and larger-scale (for entire sections of a work) forms.
Modulation tropes include: