"Cecilia" (Simon and Garfunkel)

"Cecilia" and I go way back.  It was my favorite track from Bridge Over Troubled Water (1970) as a kid.  Its all-too-brief 2:55 would fly by, and I'd want to hear it again.  And again.  My mom was always reluctant to replay it, though—no doubt because of the lyric Making love in the afternoon with Cecilia, up in my bedroom...  But I had no clue at the time what Paul and Artie were singing about.  I just liked the melody and that crazy percussion.  It made me happy.  Still does.
The song began as an impromptu jam.  The duo was having a party at a rented house in the Hollywood Hills.  At some point, they started patting out this syncopated rhythm on their thighs, and then others joined in—playing the seat cushion of a piano bench, strumming the slackened strings of an old guitar, etc.  They liked the groove, so they put it to tape, using a home reel-to-reel recorder with a slap-back reverb feature (which is what gives the track its distinctive, ping-pongy ricochet sound).
Ultimately, they brought the tape to producer Roy Halee and asked him to make a loop out of a minute-long section that they thought had a particularly good feel.  By splicing sections of tape together, Halee turned one minute of tape into a literal three-minute loop.  Simon quickly came up with the melody and lyric about a guy whose girlfriend cheats on him but then comes back to him in the end.
"Cecilia."
The older I get, the less I think the song is about a flaky girlfriend, however.  At least not in the literal sense.  It's about music and songwriting.  
Cecilia is the patron saint of music and musicians.  Putting it in ancient Greek terms, she's a muse.  And as anyone who has tried their hand at songwriting knows, sometimes the muse is kind and makes you feel like the ideas are never going to stop flowing.  Other times, she's a fickle slut who comes and goes as she pleases.  In fact, sometimes she doesn't show up at all, leaving your self confidence in the gutter.  But when she comes back, and you're suddenly firing on all cylinders again, somehow, it's even better than the first time around.  All is forgiven and forgotten.  Jubilation!  
(That Paul Simon is a clever one.)

"The Boxer" (Simon and Garfunkel)

"The Boxer" is one of those songs that I've pretty much known since infancy.  Bridge Over Troubled Water (1970) made at least a monthly appearance in our household, and I always looked forward to this track (side 2, band 1) every time it was on the turntable.  Even though the verses were much too complex and poetic for me to grasp as a toddler, I had no problem with the easy-singalong lie-la-lie vocals in the chorus.  In fact, I remember feeling a certain ownership of the song.  It was as if by knowing that chorus, the song had become mine somehow.  I've thought about that a lot over the years: how we all take possession of others' songs and make them part of the fabric of who we are.
Paul Simon wrote "The Boxer" in 1968, almost a year and a half before Bridge Over Troubled Water was released.  Lyrically, it draws from the immigrant experience: the promise of success pitted against the harsh reality of survival, where your only real choice is to persevere or die.  Likewise, it draws from Simon's weariness with his detractors.  He and Art Garfunkel had gone from critical darlings to so-called "sellouts" and "squares" in a matter of a few short years.  The imagery of a boxer getting pummeled but not giving up the fight was a metaphor for Simon himself as an artist.
From a musical standpoint, it's also quite the composition.  And quite the recording.  It begins simply and sedately enough, with Simon fingerpicking along with guest guitarist Fred Carter, Jr.  But by the time you reach the first chorus, where the melody suddenly dips into a minor key, reflecting the self doubt and perils of the protagonist, you get a foretaste of the epic crescendo that occurs in the final chorus.
And the sound of that final chorus owes a lot to the expertise of Simon and Garfunkel's engineer/co-producer, Roy Halee.  For example, it was Halee who decided to record session drummer Hal Blaine's kit in the hallway of Columbia studios, right in front of the elevator bank, to get that huge, shotgun-like boom that erupts after each lie-la-lie.  
(In the 2011 documentary The Harmony Game, which explores the making of Bridge Over Troubled Water, Halee and Blaine tell an anecdote about an elderly security guard who'd come down to check on the commotion.  Says Blaine, "I was smashing these two drums—humongous explosion.  All in perfect synchronization as my hands were coming down, the elevator door opened. There was probably an 85-year-old security guard standing there, who thought he’d just been killed.")
There are more subtle touches on the track that Halee concocted, too.  For instance, one of my favorite moments comes after the third verse, where there's this little Bach-like instrumental interlude.  For years, I'd assumed it was a Moog synthesizer playing the majestic-sounding line.  Turns out (as Simon recounts in the aforementioned documentary), Halee paired a piccolo trumpet with a pedal steel guitar, played without the attack for a more violin-like sound, and then mixed them down to sound like a single instrument.  Stroke of genius.
Anyway, it's a song that I can never listen to just once.  I usually have to play it one time to soak in the words, another time to soak in the instrumentation, and a third time to absorb it all together.  And it really is more than just a song; it's the unbridled creativity and the ugly brutality of the American experience, set to music.

"The Only Living Boy in New York" (Simon & Garfunkel)

It's early 1969.  Art Garfunkel is in Mexico, trying his hand at acting in a film adaptation of Joseph Heller's Catch-22.  Filming is taking months longer than anticipated, and Paul Simon is back in New York City, trying to write new music for their upcoming album without his best friend to bounce ideas off of...
In Jennifer Lebeau's 2011 documentary The Harmony Game, which explores the making of the album Bridge Over Troubled Water (1970), Simon discusses the gestation of "The Only Living Boy in New York," noting that the "Tom" in the lyrics is actually Garfunkel (their stage names having been "Tom & Jerry" when they were teenagers, just starting out).  
In short, the lyrics are Simon wishing Garfunkel well on his acting debut (I know your part'll go fine), while also feeling abandoned, isolated, and slighted by a lack of communication (I can gather all the news I need on the weather report).  But, ultimately, it's a painful acknowledgment that they were drifting apart (Half of the time we're gone, and we don't know where / And we don't know where), both as friends and creative collaborators.
Despite the song being a virtual farewell to Garfunkel, it's also one of their warmest vocal collaborations on tape.  Their multi-tracked, sanctuary-echo aaahhh harmonies throughout are angelically uplifting and heartbreaking at the same time.
And then there's lauded session musician Joe Osborn's bassline--a living, breathing thing that glides and plucks at all the right moments.  It's one of the key elements, along with Hal Blaine's strategic bursts of percussion, that give the song an almost Pet Sounds quality (which, incidentally, Blaine appeared on).