A-Punk

I had no idea what the song meant at first, so I had to rely heavily on others for this. Let's break it down:

"Johanna drove slowly into the city The Hudson River all filled with snow She spied the ring on His Honor's finger Oh-oh-oh"

The first couple lines are irrelevant except it means that she is visiting His Honor, so he must not be a husband. In the third line, it says she spies a ring. This could be a family heirloom or a symbol of her family. It could be a non-family member, but based on the fact that she gets the ring (more on that in a second) it's likely.

"A thousand years in one piece of silver She took it from his lily-white hand Showed no fear--she'd seen the thing In the young men's wing at Sloan-Kettering"

Here, the first line tells us that the ring is storied and old, which means it has been handed down generation to generation (or, less likely, sold). She then takes the ring and apparently inherits it, which is why His Honor is likely family. The second two lines refer not to the ring, but to death. She was unfazed because she'd seen a young man (perhaps a brother) die of cancer at Sloan-Kettering, a cancer centre. These lines don't refer to His Honor, however, because it says she'd, as in she had, so it must be before His Honor died.

"His Honor drove southward seeking exotica Down to the pueblo huts of New Mexico Cut his teeth on turquoise harmonicas Oh-oh-oh "

This part seems to be a sad flashback to His Honor seeking a last bit of fun before he died. You can tell it's from a disease because of "cut his teeth on turquoise harmonicas", which is something nobody does unless they're frail and weak. The disease could well be cancer.

"I saw Johanna down in the subway She took an apartment in Washington Heights Half of the ring lies here with me But the other half's at the bottom of the sea"

The first two lines are inconsequential except for the fact that Columbia University, where Vampire Weekend formed, is in Washington Heights, so it's implying that she moved near the singer for comfort. The other two lines, however, are more important. She must have cut half of the ring off and thrown it out to sea out of anguish and grief. She wanted to forget, so she got rid of the memories. But, at the same time she couldn't abandon her family so she gave the other half to the narrator for safe keeping.

"Look outside at the raincoats coming, say 'oh'"

Perhaps the most depressing line of the song, this seemingly insane blurb is actually quite sad. The raincoats represent her gray, melancholy spirit as she grieves, depressed and endless.

Just my take on the song. Hope this is useful and you understand it more now.