Daniel, that’s a tailor-made question for someone of my (and your parents’) generation, who came of age during the so-called British Invasion of the 1960s.  My experience of popular music has a narrow window (1967-1969), but for those three years, I was wildly enthusiastic about it.  If you want to hear what was most memorable about the music of that time, you should find one of those radio stations that plays nothing but songs from the sixties (or fifties or seventies or eighties; you probably know the nineties).  Regarding the Beatles, you need to know that “Yesterday” was once (and maybe still is) the most recorded song of all time, and that in Rolling Stone’s list of the 500 best rock albums (which came out in 2003), the Beatles had four albums listed in the top ten (including the number one album, Sgt. Pepper’s Lonely Hearts Club Band).  No other band has been as influential, but there are others whose music you ought to know, at least to some extent.  Among the original British Invasion groups, I’d include the Rolling Stones, the Who, the Yardbirds, and (maybe) the Moody Blues; among U.S. bands of the sixties, the Beach Boys, the Byrds, the Doors, the Jefferson Airplane, and (for their first two albums), Country Joe and the Fish (the epitome of acid rock).  Two late-sixties “supergroups” that made lasting impressions were Cream (from England) and Crosby, Stills, and Nash (later, Crosby, Stills, Nash, and Young).  The most important solo performer of the sixties (and one of the most influential musicians of the century) was Bob Dylan, and the best duo was Simon and Garfunkel.  Finally, I’ll add three more bands to the list (even though I don’t know anything about their music) because of their importance in underground (i.e. not top-40) rock: the Grateful Dead, Pink Floyd, and the Velvet Underground.  That should take you up to 1970.  I’ll let someone else do the next two decades. — Uncle Lee

Good grief!  I forgot Jimi Hendrix.  And as April points out, Janis Joplin.  She made only a couple of albums (I just realized how anachronistic that term is) because she died young, the same year, I believe, as Hendrix and Jim Morrison (of the Doors).  Her band was called Big Brother and the Holding Company.

Incidentally, the biggest influence on the Beatles (early Beatles, that is) was Buddy Holly and the Crickets.  They (most of them) died in a plane crash in 1959.

@csemsen - Christy (and Daniel, since you asked, too): Good question.  Most of the great bands of the late sixties didn’t survive into the seventies (the Rolling Stones were one notable exception, but I was never much of a Stones fan), and so much of what I liked about rock had begun to disappear by 1970.  At the same time, I discovered classical music.  I’m still discovering it, and I know I’ll never tire of it.

Kristi, above, gives a nice overview of the Beatles and their music.  I’ll add a couple of comments: one, that although Lennon and McCartney always took joint credit for their compositions, John was the lyricist and Paul the melodist (not always, but most of the time).  Two, that the arrangements of the Beatles’ songs were almost as important to their success as the songs themselves.  George Martin, the Beatles’ producer, was responsible for scoring and arranging the Beatle’s studio recordings, and for a time he was just as influential as Lennon and McCartney.  A few examples of his work: the strings in “Eleanor Rigby,” the baroque trumpet in “Penny Lane,” the sitar in “Norwegian Wood.”

Incidentally, there’s much more to 20-century popular music than rock and roll.  In the teens, there was ragtime (Scott Joplin was the preeminent composer in that style); in the twenties, Dixieland and the beginnings of jazz; in the thirties and forties, the big bands (Glenn Miller, Tommy Dorsey, Woody Herman, Artie Shaw: your grandparents’ music).  All of it (the best of it, that is) is worth a listen.