I use organizations a lot in my worlds. They are there to either represent the movers and the shakers the players have to work with, appease, join, and take on. They can help drive the plot and bonus when players either want to create their own powerful group or work their way up the ranks of another to obtain control. It is a tool for me as the DM to make the world seem real, and to create bonds for my players with said world.
I also find that a well built organization makes for an excellent villain. Easy to make world threatening, and hard to kill. No more 'Oh crap' moments when your players do something unexpected and murder your recurring baddie. (I ran into that a couple times as a young DM, worst feeling in the world

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As a player, I think the most successful organization encountered was called, "The White Oath". A group of powerful individuals who protected the prime material plane from outsider intervention. All intervention that it found. Whether that be demon or angel, and anything in between. Basically Outsider hunters. They would not harm those that were just being, but those that influenced or impacted the world in a major way. The group was eventually dissolved, due to some messy business with a celestial. They were not particularly liked, but they were needed and helpful in their own way.
We learned about it in passing while looting an old library. The DM at the time was just using a name generator to make some random books to come across while we searched for the information we cam for. When we asked him about the book, he basically made up the group on the spot. For whatever reason, we decided then and there to revive that group, which we did. Sought out a lot of outsiders meddling with our plane's affairs. Banishing both good and bad outsiders, those that could not be persuaded to stop what they were doing. We received a lot of praise, and some disdain. Typically though we were a force for good. It was our group though, and I will always remember that campaign and those characters and we wrestled with the tenets we all swore to uphold, even at the cost of making ourselves look like the bad guys.