Initially, it was "My Heart Will Go On" from Titanic since we had established a long distance relationship while I was in medical school in Wisconsin and he was in Atlanta.
We had some difficulties, as any couple does, during the early part of our relationship. When we finally sorted through those hard times, and he gave me a promise ring as I entered my internship year, our song became "Con Los Anos Que Me Quedan" which talks about basically spending the rest of your life making up for the difficulties which were caused by immaturity or pride. This was our first dance at our wedding to this song.
Since our wedding, I can't say that we really have a song now. Or, at least, not one that we identify with. So, I like Taylor Swift's "Our Song" because it talks about finding your song in the daily things that happen. And, it's those little things that continue to build the relationship. "...'fore I said "Amen" asking God if He could play it again."