Hey, I usually play this on piano but I translated it onto the guitar with not much difficulty, so it's note for 
note what is being played in the right hand of the piano (the principle theme, I suppose.)

Enjoy. Beautiful song as always from The National

MAIN RIFF (verses)

e|--------7------------------|
B|-----------10-8------------|
G|-(9)--9--9-------9-7-9-9---|
D|---------------------------|
A|---------------------------|
E|---------------------------|
The first note is only played at the beginning of the song, as a sort of upbeat.
Play this through the whole verse

Chorus
e|---7--7------8---8------7---7----5--5-----|
B|------------------------------------------|
G|-9--9---9--9---9---9--7---7---7-7--7--7---|  X4 
D|------------------------------------------|
A|------------------------------------------|
E|------------------------------------------|

e|-----5--5---------|
B|------------------|
G|-7-7--7--7--------|  
D|------------------|
A|------------------|
E|------------------|

The rhythm is quite hard to grasp at the beginning, especially the chorus. It's syncopated so the best way to 
get it is to listen closely to the recording
Enjoy:)