During Pinkie's surprise party, one pony in the background for some reason had Fish Eyes. The fandom nicknamed her Derpy Hooves and the name stuck, with even Lauren Faust using the name on occasion.
The mint-green unicorn that happily waves to Twilight as she heads to Canterlot's library is later seen at the Summer Sun Celebration hopping up & down excitedly while staring at the camera and standing next to a pony that looks like Bon Bon from My Little Pony Tales. The fans dubbed the unicorn Lyra, she's become almost as popular as Derpy and these were only the first of her many Funny Background Events.
Hilarious in Hindsight: Twilight's observation that "All the ponies in this town are crazy!" turns out hilarious when taking into account that every Mane Character, even Twilight, suffers a nervous breakdown at some point over the course of the season. It's made even funnier when The Hub uses the voice clip in commercials for the show; it's not false advertising!
Princess Luna became Nightmare Moon because she felt unloved. Between her and her sister, she is the more popular pony with the fans.
Jerkass Woobie: Princess Luna/Nightmare Moon. Her life wasn't particularly happy.