Since almost everypony likes social networking sites it was a given that the talented Brony fandom would produce some of their own, and indeed it has no shortage of them. If I recall correctly Flankbook was one of the original websites created for this purpose, although after a couple months it was shut down for various reasons. In the meantime several other fairly prominent websites have arisen such as the RainbowDash Network, BronyNet, BronySquare and Cloudsdale (the latter of which is more of an instant messaging client and also features mobile apps), but Flankbook's developers talked of remaking their website and launching it again for the world to use. Their minimalistic, Boostrap-based website entered open beta today, with more features on the way shortly.
On a related note, rival website BronyNet has recently released a massive update to their custom NewsTime software, which means that it now features realtime updating, profile headers and other fancy stuff like that. It currently offers a lot more than Flankbook and has a less-cluttered interface than BronySquare, but since the former is still in open beta things could change quickly. We also have a BronyNet page at http://bronynet.com/main.php#!/profile-3175, in case you are interested in following us over there for site updates and whatnot.