If someone re-imagined Westeros as a high school drama—Seven Kingdoms High—then Renly would be the most popular guy in school. Everyone would want to sign his yearbook, be in his clique, and he would throw the most killer party. What can we say? The dude's got style for miles.
Sure, the political game of the Seven Kingdoms is not a high school, but in a way, it still follows high school rules. You know what we're talking about.
Although hugely popular, Renly is also a bit of social black sheep and his personality is very unusual given the gender norms of the Seven Kingdoms.
Cressen notes that when Renly was growing up, the boy "loved bright colors and rich fabrics, and he had loved his games as well" (1.Prologue.11). These games don't include fighting in mock battles and pretending to be a warrior, but rather playing as dragons and wizards and rain gods. Renly had a very fanciful imagination it seems.
We see Renly didn't grow out of his love of color and pageantry either, as evidenced by the parties he throws on his way to war with the Lannisters. This is very different from the gender norms we witness in his brother Robert Baratheon, who enjoyed fighting, killing, drinking, and grunting.
Since he is so different than the other guys, perhaps this leads to Renly being able to openly question the feudalist society in which he lives. When Catelyn points out that Stannis is the elder brother and therefore the one with the legitimate claim to the throne, Renly replies:
"Though it's a fool's law, wouldn't you agree? Why the oldest son, and not the best-fitted? The crown will suit me, as it never suited Robert and would not suit Stannis. I have it in me to be a great king, […]." (32.Catelyn.99)
Sure, Renly has the most to gain from questioning this aspect of his society, but it still makes him a unique character—few other characters openly critique the social norms and laws of their feudalist society. Eddard Stark didn't. Stannis Baratheon sure as heck doesn't. Even Davos rolls with it. But Renly openly considers whether there is a better way to do things, making him progressive by Seven Kingdoms standards.
Given this, it's pretty tragic that Renly is assassinated before he has a chance to do much. Maybe the Seven Kingdoms just weren't ready for something completely different.
In the HBO series, Renly is straight-up gay, and Loras Tyrell is his paramour. In the novels, it's way more ambiguous, but there are several hints that Renly is, in fact, not a straight dude.
When Stannis and Renly are squabbling, Renly notes that Stannis refused to come to his wedding, but that his brother shouldn't worry as Margaery came to his bed a virgin. Stannis retorts that the wedding was a "mummer's farce" (32.Catelyn.81, 83) and that in Renly's bed Margaery would likely die a virgin. This is the only hint to Renly's sexual orientation in A Clash of Kings, but careful readers will notice subtle—and a few not-so subtle—hints dropped in later volumes of the series. Not to get all spoiler-y on you.