We have a lot of them. We've been together for more than 6 years, so really I feel like there are ones that represent different phases. :-P
Some that stick out are 'True Companion' by Marc Cohn, 'Pressure Suit' by Aqualung, and and 'Love Song for No one' and 'St. Patrick's Day' by John Mayer. Actually, really most of John's first album, because we were listening to it when we first started talking.
When it was time to pick a first dance song for our wedding, we didn't pick one we already knew, but one I had found- and I knew him well enough to know he'd agree to dance to it :-P. It was 'Stay with You' by John Legend.