Here's a short playlist by some of the imho most noticeable artists of this genre.
While making punk rock with his band Millencolin, his solo albums are full of '60s tingled singer/songwriter songs, here are couple of them:
The sometimes Beatles-esque and indie-rock-esque songs by Matt have a huge stylstic variety from '60s pop, to indie rock, folk, bluegrass or pure country.
Here are a few examples:
Former shoegazer (Slowdive) nowaday makes quiet, dreamy, peaceful folk tunes.
Iron & Wine
Best known for soundtrack contributions like "Such Great Heights" ("Garden State") or "Flightless Bird" ("Twilight") Iron & Wine, mainly consisting of Sam Beam, and his sister sarah sometimes singing background vocals, play mesmerizing songs ranging from dreamy quite acoustic tunes mainly consisting of Sam's quite voice and decent fingerpicking to '60s like experimental pop songs like on their record "Kiss Each Other Clean."
Another band by Neil Halstead, together with Ian McCutecheon and Rachel Goswell - two of his former bandmates in Slowdive. Rather countriesque - check it out bro ;-)
Simon & Garfunkel
The last band isn't exactly indie-folk but had certainly a huge impact on this genre, and can be seen as some kind of "fathers" of this sound. Even though you might think, that they suck, because finding them in the record collection of your parents, you should give it a try, and listen to same rather unknown but unique songs, such as:
Hope you had fun and digged a few new tunes, this list doesn't intend to be an all embracing overview over the genre, just a selection tinged by my personal taste of music. For completeness, there are lots of other bands out there playing nice indie folk music, such as: Mason Jennings, Conor Oberst, Bright Eyes, Of Monsters And Men, Bon Iver, William Fitzsimmons, The Decemberists etc.