There is no single governing body of Freemasonry in the world. The United Grand Lodge of England (UGLE) is the descendant of the first Grand Lodge formed in 1717, but that gives it no authority over other Grand Lodges, all of which are equal. The UGLE does not even have total authority in Great Britain, for Scotland has its own Grand Lodge.
The Supreme Council of the Ancient And Accepted Scottish Rite for the Southern Jurisdiction of the USA, sometimes is called the Mother Supreme Council of the World, for it was the first to be formed, but again, all Supreme Councils are equal, and chronological primacy confers no special authority. The Southern Jurisdiction of the AASR does not even have complete authority in the USA, for there is also a Supreme Council for the Northern Jurisdiction, comprising the states east of the Mississippi and north of the Ohio River and Mason-Dixon Line.
There are the General Grand Chapter of Royal Arch Masons International and the General Grand Council of Cryptic Masons International. But these bodies have mostly ceremonial impact; no Grand Chapter or Grand Council is required to belong to its General Grand counterpart, and many Grand Chapters/Councils do not.
Finally, the top authority in Masonry is always the Grand Master of Masons, not some Grand Commander or other personage associated with the "higher" degrees. The Grand Master of Masons can suspend the General Grand High Priest from all the privileges of Masonry; the GGHP has no such power. Obviously, there is no global organization in Masonry.
Conspiracy
The most bizarre thing about conspiracy theories in general is that there is never a clear explanation of what the conspiracy is about, nor how it is carrying out its aims. The alleged Masonic conspiracy stories conform to this. None of the conspiracy theorists ever explains what it is that the Masons want to do with their supposed power.
Since Masonry's tenets are brotherly love, relief, and truth, if the Masons did run the world, it might be a better place. Many of the Founding Fathers who wrote the Constitution of the United States were Freemasons; the principles in that document have stood the test of over two centuries. Would a Masonic government be so bad? Look at the governments founded by anti-Masonic groups: Nazi Germany, the Soviet Union, Iran under totalitarian religious rule. Where is the real problem in the world?
Bavarian Illuminati
This group died out in the 18th century. An organization that does not exist is a convenient scapegoat! To the conspiracy loony, that there is no evidence of a group's existence is "proof" that it is fiendishly clever in concealing itself. One does not have to be a professor of philosophy to see that this kind of logic makes no sense in a search for truth.
Masonic symbols on the dollar bill
Some commentators have claimed that there are Masonic symbols on the US $1 bill, and that they were put there by the Masonic president, Franklin Delano Roosevelt to show that the country had been taken over by Masons.
Well, perhaps the symbols are Masonic, but the material on the dollar bill dates from the late 1700s, not Roosevelt's term. The two circled objects on the back of the bill are the two sides of the Great Seal of the United States. It is said that Ben Franklin, a Mason, had some influence in the design.
What are these Masonic symbols? The representation of an eye and an unfinished pyramid. The All-Seeing Eye of Deity is certainly mentioned in Freemasonry, but that concept dates back to the Bible, at least. An unfinished pyramid symbolizes that the work of nation building is not completed, but the pyramid is not a particularly Masonic symbol; any unfinished building would have done. (Some say that there is an owl in the engravings in one corner of the bill, but that is a product of an overactive imagination. The owl is also not a Masonic symbol; the only birds that come to mind in any of the degrees are the pelican in the 18th degree [a symbol of Jesus, incidentally], the mythical phoenix, and the eagle. And those are found only in the Scottish Rite, so they are not characteristic of Masonry as a whole.
New World Order
Ever since George Bush (not a Freemason) publicized this term, it has been an obsession of certain groups. They point to the wording on the dollar bill (see above), which reads "novus ordo seclorum." Unfortunately, as someone once said, "Th[eir] Latin waxeth rusty." The phrase on the bill means "a new order of the ages," and refers to the completely novel (and still unique) form of American government, a republic of separated powers, composed of a federal union of states, in which the central government is granted powers by the people, whose rights are supreme over the institutions of government. If the term were to mean "new world order," the third word would have to be "sæculorum" instead.
The Kennedy Assassination (and others)
Much has been made of the facts that many members of the Warren Commission were Freemasons. Supposedly, this allowed them to "cover up" the "evidence" that the Freemasons had Kennedy assassinated. Of course, there is no explanation of how the Freemasons might have benefited from Kennedy's death or what other motivation they might have had for such a plot. For most of the history of the American Republic, about one-third of American officeholders--presidents, senators, judges, congressmen, local officials--have been Freemasons. It is hardly surprising that a group such as the Warren Commission would have been about 1/3 Freemasons.
As for other sensational assassinations, there is the same question to be asked: How could the Freemasons have benefited from this act? As there is never a sensible answer, the allegations are clearly laughable.
Since the Freemasons have been around for nearly 300 years and have held many responsible positions in the American government, as well as in other countries around the world, particularly the English-speaking ones, if there were any such conspiracy, it would have long since succeeded in its aims. As the concept is the product of overwrought imaginations, the total lack of evidence or purpose for any such conspiracy must lead us to dismiss it as nonsense.
You said that Masonry was not a religion and had no priests, but you just mentioned a Grand High Priest--what gives?
At the time of the return from the Babylonian exile, some of the legendary events of which are commemorated by the Royal Arch Degree, Jeshua, Zerubbabel, and Haggai were the High Priest, King, and Scribe among the Israelites. The important roles played by these individuals led to their positions being used to designate the three principal officers of the Royal Arch Chapter. The title "High Priest" is used by the presiding officer of a Royal Arch Chapter in the United States of America. In other countries, the title King is assigned to the presiding officer, and the High Priest is a subordinate officer; anti-monarchist sentiment in the US at the time the Royal Arch degrees were becoming established in America (late 1700s) led to the primary role being assigned to the High Priest, which was also consonant with the American notion of the state being subordinate to the Deity. In some foreign jurisdictions, the designations of the first three officers are simply First, Second, and Third Principal, with no reference to the historical roles of the individuals commemorated in the degree ceremonies.
The High Priest of a Royal Arch Chapter (and likewise the Grand High Priest of a Grand Chapter and the General Grand High Priest of the General Grand Chapter) performs no sacerdotal functions; his office is the equivalent of those of Worshipful Master of a Lodge, Illustrious Master of a Council, Eminent Commander of a Commandery, and so on. Like all Masonic bodies, a Royal Arch Chapter has a Chaplain (in Scottish Rite Masonry, the office is termed "Prelate") who is responsible for offering prayer at the opening of a meeting. The Excellent High Priest is responsible for administration of his Chapter's business and conduct of its ritual.
Though it has Commanders, Masonry has no army; though it has officers titled "High Priest," Masonry is not a religion. The High Priest is not a priest, paradoxical as that may sound; he is a chairman or president in fact, if not in name.