*Days of Future Past Spoilers* (If you haven't seen it, you need to.  It was an awesome movie.
Named for their founder, and one of the most famous Mutants ever, they have manipulated events for thousands of years to get humans ready to live peaceably with Mutants.  In the past the humans either idolized you, as prophets, gods, or demi-gods, or hated you, believing you were a demon.  The Order, however, was founded on the idea that humans will learn to accept mutants as equals, leading to true coexistence for both groups.  They have done their best to remain hidden from the world, from both mutants and non-mutants, and did their best to keep the existence of mutants from being common knowledge.  After WWII they started to wonder if humans might be ready to learn of the existence of mutants, but they didn't have to wonder long, as in 1973 mutants were caught on camera at the treaty signing in Paris, and days later the same mutants were caught trying to murder President Nixon.  When one of them saved his life, however, the first link in the chain of trust was struck.  Humans began to see that mutants weren't evil, that they could be quite good.
The peace wasn't easy won, though.  The Order must work constantly to maintain this peace, and when so many lives hang in the balance, there are few things that are considered "going too far."
When, occasionally, someone discovers the existence of the Order, they are given a choice.  Join, or be given amnesia so that you are no longer a threat.

I want Mutants, but in an organization like this, normal humans would be useful as well.  It occurs in the new timeline from the end of XM:DOFP, and as such, XMO:W, XMFC, and the history and alternate future from XM:DOFP are the only cannon parts from the films.  We can make up the rest.  Such as:
Mystique is now Senator Raven Darkholme from NY.  After saving the president's life, she worked for the CIA as a covert operative until the end of the Cold War.  She then went to law school and got a job with the ACLU specializing in mutant rights.  When she ran for the Senate 7 years ago she got 61% of the vote, bipartisan support because she ran as an independent.