Birthday

"Birthday" - written by John and Paul, and featured in "The Beatles" (aka White Album). Not to be mistaken with "Happy Birthday", which is a cover of the traditional birthday song by the Beatles.
This song is also a link that is often included in birthday wishes in Facebook. So if someone sent you a link to this page, "Happy Birthday!" to you. :D

I've Got A Feeling

This song is a mashup of Paul McCartney's "I've Got a Feeling" and John Lennon's "Everybody Had a Hard Year" from the White Album sessions. Featured in the "Let It Be" album from 1970.
Lennon's looked very kewl with the guitar in this video.

Two Of Us

"Two of Us" written by Paul McCartney and sung as a duet by Lennon and McCartney. In the "Let It Be" album, released in 1970. George Harrison played bass in this song. Poor Ringo seemed to be bored to death playing drums here. :(

Hello, Goodbye

"Hello, Goodbye" from 1967. Video features a light hearted John Lennon dancing towards the end of the video. Now contrast this with his 1969 videos where he looked so serious. What happened in those 2 years?

Hey Bulldog

Beatles recording Hey Bulldog, a song that started out as Hey Bullfrog until someone (Paul?) started making dog sounds towards the end of the song. :)

Get Back

Beatles Rooftop Concert at Apple Corps (NOT Apple Computers!), 30th Jan 1969. This was their first live performance in public since they stopped touring as a band years earlier. And 1 year before they broke up as a band.

Come Together

You can hear John Lennon saying "Shoot me! Shoot me! Shoot me!" over and over again in this song from Abbey Road, 1969. That aside, "Come Together" is, for me, one of the most creative and inspired Beatles songs.
This song was recently (April 2010) covered by Crystal Bowersox in American Idol during the Lennon-McCartney Songbook week.

Revolution

"Revolution" is an anti-establishment song written and sung by John Lennon. The song, though about wanting a revolution, is against achieving it through violence though. From the B-side of the "Hey Jude" single, released in 1968.
Note the "shoo-bee-do-wah" that was removed from the single in this particular version.