Organizations are large groups in the city that the player can choose to work for, so any organization that is willing to talk to your faction is available for you to engage with. Each organization is made up of contacts in a kind of hierarchy.
The contacts you deal with at the start are low level guys, and the missions and things that you are doing are more goffer tasks and street level stuff for the organization. The higher you go up the more you start dealing with more important contacts in that organization, and eventually if you do well playing the game, you'll work for like the head of that organization. At that point you start dealing with more in-depth, more managerial style things; you're more into the guts of the organization and doing the more important stuff that they can only trust the more trusted individuals to do.