Their relationship: Logan Huntzberger is the dream: the playboy who reforms his ways when he meets the one girl he just can't live without. He's perhaps the most unlikely long-term relationship for Rory — she can't stand him when she first meets him. We can't really blame her; Logan's privileged, spoiled, and arrogant . . . he just also happens to be a lot of fun.
Why the relationship is important: Throughout their (rather tumultuous) relationship, both Rory and Logan do a lot of growing up. They're very different people, but they always come back to the conclusion that they care about each other too much to call it quits. By the time Gilmore Girls ended, Logan had himself a stable career and was ready to marry Rory. But his proposal is what shows Rory that she needs to pursue her own dreams before settling down. She chooses to live her own life in her own way over being with the guy she loves, and that's gutsy.