For those who don't know, the concept was thought up by Hamster quite some time ago. He thought of a group of players wearing only greyscale joints and textures, creating a sort of style movement or uniform within Toribash.
He eventually got round to posting BnW, and a load of other "orgs" literally sprung up overnight.
MahaTmA is correct; an organisation is (in this context) nothing but another way of bringing players together. The difference is that organisations appear to have more original requirements (or none at all), such as joint colour. As organisations tend to lack the structure of clans, there isn't a great deal of actual comradery going on ingame, or, at least, little that I have seen.
In my opinion, organisations have become a fad. They'll fall apart at some point.