Their relationship: The big one. Jess Mariano is reckless, irresponsible, and impulsive when he arrives in Stars Hollow in season two and is generally pretty rude to almost everyone in town except Rory herself. Honestly, that would be hard for any girl to resist. Once Rory and Jess actually start dating, she comes to realize that Jess is just too angsty and closed off to make a real relationship work. And come on, leaving town for California and not even telling Rory? That's cold.
Why he's important: Jess is wrong for Rory in many ways at the time she dates him. But he then matures into someone who could be just right . . . just at the wrong time. By the time Jess gets his act together and gets back in contact with Rory in season four, she's on her way to getting back together with Dean. And then when they meet again in season six, she's with Logan. Jess is Rory's ultimate "what if?" guy, and that's what makes him so intriguing.