A brony is any person outside of the target demographic who thinks that My Little Pony: Friendship is Magic is a great cartoon.
Some people will say that "bronies" are only adult males, but there are lots of female bronies out there who are proud to be called bronies and wouldn't want any other title! Some may call themselves pegasisters, or perhaps fillies, and I've even heard of "bronettes" once, but in the end we're all united under one pony-shaped umbrella: we are all eagerly awaiting the next episode of MLP:FIM to come out on Saturday! (and maybe it'll have a song, please have a song, oh please oh please)
Why My Little Pony? That's not really the right question; it's more like... why not? Lauren Faust, famous for her heavy involvement with The Powerpuff Girls and Foster's Home for Imaginary Friends, gave her whole creative being into making a show that exemplifies her opinion that entertainment aimed at girls doesn't have to be forgettable chaff. She was able to pull together an incredible staff of passionate animators and story tellers to make something completely unforgettable. This cartoon has it all: characters, comedy, animation, music, and, yes, ponies, and we wouldn't have it any other way!
Curious? Good! Equestria Daily is one of the major hubs of the fandom, and a great place to start poking around, and with some casual searches on Youtube you should be able to find all of the episodes available for streaming. And once you're hooked be sure to support the show by purchasing official merchandise and spreading the pony fever to your friends! You're shaking your head now, but... once you have a favorite pony you'll know you're sunk.