I love the sounds of artists/bands like Sufjan Stevens, Andrew Bird, St. Vincent, Grizzly Bear, Anathallo, etc..sort of has a modern classical feel to it while still being indie. Live performances by these guys are usually incredible. not exactly the same but Beirut is also amazing, gypsy flavored folk music.
Edit: oh, I was supposed to list albums here. These artists have multiple albums so i'll list my favorites
Sufjan Stevens - Illinois. Should be one of the starting points of those looking to get into indie/folk/whatever music. And clocking in at well over an hour, it's pretty epic, too. The album itself is dedicated to the state, Illinois if you couldn't tell.
Andrew Bird - Noble Beast. I compare him to Sufjan a lot although Bird's sound is more rocky at times. I've actually not yet to see him live because tickets are expensive yet sell out within hours/days.
St. Vincent - I can't pick a favorite album from her. Get all 3: in chronological order it goes, Marry Me, Actor, and Strange Mercy. Marry Me was a solid, adventurous debut but I feel like she hit it out of the park with Actor. Honestly she doesn't get as much credit as she should! Strange Mercy was phenomenal too; it scratched my top 3 of 2011.
Grizzly Bear is my favorite indie band. They solidified their position when I saw them live a few years ago, right after Veckatimest got released. They were on tour with Beach House (side note: Check out Beach House too!) and they blew away every single expectation of how I thought they would sound live. Overwhelming amount of clarity, the cleanest, clearest instrument separation, seriously you could have cut it with a knife. For just being four guys they worked hard to faithfully recreate the music on the record and, well, I ended up going home very disappointed because from there on out, I felt the albums
sucked compared to their live show. But oh well
Anathallo - Floating World. I like to pair this album next to Sufjan Stevens because of similar feel. Except Floating World has this whole Japanese folklore thing going for it in the lyrical content. Very enjoyable.