At the very cornerstone of songwriting are chord progressions. They are, simply, chords played in sequential order. But chord progressions aren't as easy as just knocking out a few chords in a row. Chord progressions must somehow be tied together; in order to be a true chord progression, the chords must feel like a whole. Creating this entire entity can be achieved in a number of ways; a songwriter may link chords based by proximity or by common notes. Chord progressions might even be based on how absolutely different the chords are, it really doesn't matter. It's all based on the style and taste of an individual songwriter; the best songwriters can make strange, seemingly unrelated chords into viable chord progressions.
Chord progressions are based on a series of chord changes, and these changes form the basis for the melody to be formed. Chord progressions are the harmonic backbone of a song, and they often dictate the song's tone and mood. Modern music tends to frequently base chord progressions out of the first, fourth, and fifth degrees of the scale; in C major, this would be C, F, and G. Of course, these chord progressions can be varied in a number of ways (chord substitution allows heavily for that), but these basic chord progressions tend to be the framework for a decent portion of modern music -- especially rock and pop.
There are no specific rules governing chord progressions, but modern rock and pop music seems to thrive on the most minimal amount. The early days of both rock and punk music were full of songs with only three chords in the chord progressions and only a few minor substitutions. Because punk music was the province of those shunning an overwhelming knowledge of music theory, these simple chord progressions could have been the result of an inability to play much else. But simple chord progressions survived that period and moved through to musicians who have been classically trained; this has everything to do with the catchiness of simple chord progressions. A three or four chord progression gives freedom for the melodies to be explored to far degrees and allows the listener to remember the chord progressions very easily. If the chord progressions are memorable, they tend to be repeated over and over again, creating a unmistakable catchiness. And that catchiness is what has made simple chord progressions the most desirable form in modern rock and pop.