Title basically says it all. I have limited experience listening to different headphones, so I'm basing my preferences off of the few I own and have heard in the past.
My trusty sportapros have been my knockaround gym and work phones for a few years now. I don't think they sounded this great when I first bought them, but I listened to quite a bit of pink and brown noise while studying for school in the past year, and now they sound fantastic (or I've given in to their sonic signature and just gotten used to them). My ATH-M50's sound even better. I can't articulate exactly what it is about these phones, but they both have a certain clarity that I really enjoy, without sacrificing impact, if that makes sense. The M50s give acoustic insturments an amazing richness that I really enjoy, and they're articulate without being cold in any way.
All that said, I was given a pair of Sony MDR-X05's that don't suit me. While they are the most comfortable pair of phones I've ever worn (with killer isolation to boot), the bass is overwhelming to the point where vocals and guitars feel distant and somewhat obscured. As a guy who listens to a lot of rock based genres of different extremes (folk and jangle pop on one end, death, black, and doom metal on the other) but very little dance or hip hop, I assume I'm just not a basshead.
But with the headphones that I do like... what category do they fall under? It's hard to research other types of headphones (such as IEMs) when I'm not sure which I should be looking at.