What was the Mayflower Compact?

The Mayflower Compact was a set of rules for self-governance established by English settlers heading to America on board the Mayflower. It was the first agreement for self-government to be created and enforced in America. In the compact, the pilgrims start by expressing their unswerving loyalty to King James I:

In ye name of God Amen· We whose names are vnderwriten, the loyall subjects of our dread soueraigne Lord King James by ye grace of God, of great Britaine, franc, & Ireland king, defender of ye faith, &c."

The pilgrims had unintentionally arrived at Cape Cod, miles from their destination in Virginia, due to storms and inclement weather. They now found themselves outside the boundaries of the Virginia Company patent which had been granted by the king. This led to a number of pilgrims on board the Mayflower insisting that they were no longer answerable to the terms of the original patent.
The compact was a response to the incipient rebellion, an attempt to impose some semblance of order upon an increasingly mutinous ship's company. That being the case, it was necessary for the pilgrims to begin their document with an open expression of loyalty to the king who had granted the Virginia Company their royal charter.

Haueing vndertaken, for ye glorie of God, and aduancemente of ye christian faith and honour of our king & countrie, a voyage to plant ye first colonie in ye Northerne parts of Virginia· doe by these presents solemnly & mutualy in ye presence of God, and one of another, couenant, & combine our selues togeather into a ciuill body politick; for ye our better ordering, & preseruation & fur=therance of ye ends aforesaid; and by vertue hearof, to enacte, constitute, and frame shuch just & equall lawes, ordinances, Acts, constitutions, & offices, from time to time, as shall be thought most meete & conuenient for ye generall good of ye colonie: vnto which we promise all due submission and obedience. In witnes wherof we haue herevnder subscribed our names at Cap=Codd ye ·11· of Nouember, in ye year of ye raigne of our soueraigne Lord king James of England, france, & Ireland ye eighteenth and of Scotland ye fiftie fourth. Ano: Dom ·1620·

In the remainder of the document, the pilgrims formalize their establishment of a government, a "civil body politick" to which they agree to subject themselves upon landing. The message is clear: the pilgrims may be outside the boundaries of the original legal agreement, but they are not outside the law. They have established a temporary government, and it will be a government of laws. The king did not authorize the unleashing of anarchy upon the colonies, and he can be sure that the pilgrims will behave as decent, upstanding English subjects should.
The pilgrims solemnly swear before God that they will establish a lawful government in full obedience to King James. Swearing to God gives the pilgrims' words an added force as no one could truly doubt their Christian faith. These words are chosen carefully to provide reassurance to the king that the pilgrims are genuine in their stated aim to set up a civil government on American soil under his sovereignty.
https://www.britannica.com/topic/Mayflower-Compact