I've been looking forward to including Sigur Rós on this blog, because there aren't many people I know who are aware of the band. But they're probably aware of bands who have been influenced by them, such as Radiohead and Coldplay (just to name a few). I've chosen two tracks from their album "Takk...", because I think you need both to get a feel for what they're capable of. For "Glósóli" (roughly 'glowing sun' in Icelandic), I've embedded the official video for the song, because I feel that it enhances the experience of hearing it. For "Sæglópur" (roughly 'lost at sea' in Icelandic), I've chosen to instead go with an unofficial video that's mostly a black screen, because in this case, I think the official video takes the focus too far away from the music (you can still view it here afterwards if you like, though). The band is best experienced with headphones, but with Youtube videos, the sound quality is only going to be so good, regardless of the setup.
One other thing: make sure to listen to both in their entirety, because a 30 second snippet won't give you an accurate picture of the songs. Hearing Sigur Rós for the first time can be a bit of a trip, so I sincerely hope you enjoy both of these songs!
Note: Massive props to Craig Ratliff for pointing me to Character Map, without which this post would not have been possible...Crazy Icelandic.