Typically, the three chords used are the chords on the tonic, subdominant, and dominant (scale degrees I, IV and V): in the key of C, these would be the C, F and G chords. Sometimes the V7 chord is used instead of V, for greater tension.
There are literally tens of thousands of songs written with I, IV and V chords. Almost all country, blues, and early rock and roll songs are three chord songs. A great many pop songs also are I, IV and V chord songs.
In the mid-1960s, two of the most popular bands, The Beach Boys and The Beatles, began releasing songs that stretched the scope of rock and roll beyond three-chord songs. Even their earlier hits, such as "The Warmth of the Sun, or "She Loves You, featured chord progressions somewhat more complex. This led to a movement away from the country and blues base of rock and roll music, towards what would be termed simply rock music, and eventually resulted in the development of progressive rock and its many derivatives. However, the popularity of the three-chord song has always remained.
Songwriter Harlan Howard once said "All you need to write a country song is three chords and the truth."
Lou Reed said "One chord is fine. Two chords is pushing it. Three chords and you're into jazz. (Reed has nevertheless written many songs with unique or complex chord progressions himself, such as the material on Berlin.)