For many listeners, "New Slang" is the Shins song. If you've only heard one of their songs, this is probably it; NPR called it "the Shins's masterpiece" (source). The song is not just a one-off success, though: it's highly typical of the Shins' early recording style and songwriting.
Most of Oh, Inverted World makes use of two specific techniques: adding reverb, and doubling the vocals. Songs like "New Slang" sound kind of echo-y, as though they were recorded a really long time ago. That's the reverb, which James Mercer added in himself when he recorded the album. (Reverb is short for "reverberation," like what you get when you talk down a long, empty hallway.) Because modern recording equipment has made it possible to get extremely crisp sound, the extra echo sounds typical of far older albums.
Mercer also doubles his voice on almost every song, because, as he explains, "I was in a little studio apartment so I always felt self-conscious about the singing because I knew my neighbors could hear" (source). Since he wanted to sing quietly, he simply recorded the vocal tracks twice to give his voice extra power in the mix.
In terms of style, the band's repertoire is full of simple songs that pack a powerful punch. They aren't simple because the band lacks musical skill, but because James Mercer created the band specifically to make short pop songs, and pop is often made up of very basic musical elements. What makes "New Slang" so definitive of the early Shins sound is that it is the essential simple pop song. It has a straightforward verse to chorus progression, and it uses only vocals, guitar, and bass with extremely basic percussion. In fact, it seems to be exactly the type of song Mercer wanted to create when he started the band.
"New Slang" is also the first single that the Shins released that saw wide distribution, because it was the first one released by the band's label, Sub Pop. Pretty much every other Shins song, whether rightly or wrongly, has to live up to this one.