I happen to be a big fan of the band "The National" and I find it interesting how polarizing this band seems to be among rock music lovers. This is truly a love/hate type band.
Some bands require an investment to "get" the music. The National requires a bigger investment than most bands in my opinion. Their music is very unusual and really defies classification. There music is at both repetitive AND complex. I find that the most satisfying art requires some time to appreciate. The National offerings are in the category that get better and better the more you hear them. Which I think is one reason that they are polarizing. It would be easy to dismiss their music after one listen. But put it on a few times (preferably with some intoxication to lubricate the mind) and it grows on you and becomes more powerful. Their albums also truly demand a complete listen. Their songs do not stand well in isolation and I don't think they are intended to. Their albums seemed to be designed to take you on a mood journey and that is destroyed by cherry picking the songs.
Anyone not familiar, I'd recommed starting with Boxer, but Alligator and High Violet are also excellent.