Derpy Hooves is the name given to a gray pegasus pony with a cutie mark of bubbles, by the brony community.
She is mostly seen with crossed, or otherwise distorted eyes, and may very well have been creator goofing around or making a mistake.
Many people have Derpy as their favorite pony, and there is even a flash game named Derpy Delivery in development.
Derpys real name may be Ditzy Doo, since it is referenced in the Winter Wrap Up episode, and was also mentioned by Lauren Faust.
There's many instances where multiple Derpy's can be seen in a single frame of the show, but the most common ones have only 1, and her eyes are derped out.
There is only 2 currently possible speech lines for Derpy, one being Muffins, and the other being a mess of hushed speaking in a crowd when Trixie, another pony from the show, is showing off.
Derpy Hooves is almost completely fan made, never the less she is one of the most popular.
The collective group of My Little Pony: Friendship is Magic fans.
They have multiple web sites dedicated to the show, along with many fanfics and fan made videos based upon the show.
The Bronies tend to be passive aggressive, hardly ever cussing people out or talking in all caps, though this is not to be said about the people who despise brony kind.
There is currently a debate between bronies and non-bronies on whether or not people should watch the show, and if it should be spread around.
Bronies can not be stopped, it will spread until one of 2 things happens.
1. Lauren Faust, the most famous worker on the show, and co make a horrible season(s).
2. The show ends.
The later will take much more time to dispose of the Brony community, but will inevitably do so unless an entirely new show is made much like how the current one has taken place of the old.
There are many popular phrases used by bronies, such as brohoof, and are common place in discussion between bronies.
Though there are many phrases bronies use and that are tolerated, there is one that most, if not all bronies loath to see, and that one quote is most commonly used by a fellow brony.
That quote being "_______ (being a pony from the show) is my waifu!"
The last thing bronies like to do with phrases is replace words in common day sentences with pony related words, such as replace everybody with everypony, or main with mane.
For any other information wanted related to Brony(ies), visit Equestria Daily, one of it's many sub pages, or the Know Your Meme page about it.
The most common games to find bronies in are Minecraft, Terraria, and League of Legends. (these are only to my current knowledge)