Doctrine of the two kingdoms
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He rules the worldly or left-hand kingdom through secular (and, though this point is often misunderstood, also churchly) government, by means of law (i.e., the sword or compulsion) and in the heavenly or righthand kingdom (his spiritual kingdom, that is, Christians insofar as they are a new creation who spontaneously and voluntarily obey) through the gospel or grace.
The two kingdoms doctrine is simply another form of the distinctive Lutheran teaching of Law and Gospel. The official book that defines Lutheranism called the "Book of Concord" compiled in 1580 references a sermon by Martin Luther on this from 1528 preached on the 9th Sunday after Trinity in Marburg, that is about the Two Kingdoms or Two Kinds of Righteousness.
In that sermon he states that the wordly (left hand) Kingdom includes everything we can see and do in our bodies. This fully and especially includes whatever is done in the church. This is taught so that it is clear that in the Heavenly (right hand) Kingdom, the only thing that is included there is alone faith in Christ. "Christ alone" and "faith alone" are Lutheran slogans that are reflected in this way.
The biblical basis for this doctrine, as with all Law and Gospel modalities is the distinction St Paul makes in Romans 8 between "flesh/body" versus "spirit/Spirit". Martin Luther's breakthrough moment was his break with the traditional scholastic understanding of this passage. The Scholastics understood flesh vs spirit to be the movement from vice to virtue, from the profane/secular/civil to the sacred/churchly.
Luther saw this contrast instead to be a movement from true virtue, which especially included the sacred and churchly and any righteousness we can do or that is visible, to alone the invisible righteousness of faith in Christ, which in the sermon referenced here he says is "meaningless on earth except to God and a troubled conscience."
On Secular Authority 
Martin Luther's book, On Secular Authority, was an ardent expression of the principle of Liberty of Conscience. "Liberty of conscience" is the principle that forbids human authorities from coercing people's spiritual beliefs. In this book, Luther insisted that God requires voluntary religious beliefs. Compelled or coerced faith is insincere and must never be allowed. Luther insisted that "liberty of conscience" was one of Jesus Christ's principles. According to Luther, the civil government's role is simply to keep outward peace in society. The civil government has no business enforcing spiritual laws. "The laws of worldly government extend no farther than to life and property and what is external upon earth," Luther insisted. Echoing Luther, writing on religious liberty, Thomas Jefferson stated "The legitimate powers of government extend to such acts only as are injurious to others." Jefferson may not have had Luther specifically in mind, but was perhaps an heir to the Protestant tradition which gave birth to this sentiment. Addressing the question of whether the state should permit its citizens to believe religious views which are heterodox, Luther said, "heresy can never be prevented by force…heresy is a spiritual matter which no iron can strike, no fire burn, no water drown." In other words, it is folly to legislate and enforce religious beliefs.
Luther's articulation of the parameters of civil government was a monumental step in the development of the separation of church and state. He argued for a clear distinction between two separate spheres: civil and spiritual. This is known as the Doctrine of the two kingdoms. The civil sphere deals with man's physical life in society as he interacts with other human beings; in this, man is subject to human governments. The spiritual sphere deals with man's soul, which is eternal, and which is subject only to God. The Doctrine of the two kingdoms is articulated by Luther in these terms:
God has ordained the two governments: the spiritual, which by the Holy Spirit under Christ makes Christians and pious people; and the secular, which restrains the unchristian and wicked so that they are obliged to keep the peace outwardly…The laws of worldly government extend no farther than to life and property and what is external upon earth. For over the soul God can and will let no one rule but himself. Therefore, where temporal power presumes to prescribe laws for the soul, it encroaches upon God's government and only misleads and destroys souls. We desire to make this so clear that every one shall grasp it, and that the princes and bishops may see what fools they are when they seek to coerce the people with their laws and commandments into believing one thing or another.
Luther forbade Christians from allowing temporal rulers to meddle with their hearts in matters of belief, declaring that "if you give into him and let him take away your faith and books, you have truly denied God". However, in all temporal matters, subjects must obey and welcome true Christian suffering:
We are to be subject to governmental power and do what it bids, as long as it does not bind our conscience but legislates only concerning outward matters…But if it invades the spiritual domain and constrains the conscience, over which God only must preside and rule, we should not obey it at all but rather lose our necks. Temporal authority and government extend no further than to matters which are external and corporeal.
Response and influence 
There are two governments: the one religious, by which the conscience is trained to piety and divine worship; the other civil, by which the individual is instructed in those duties which, as men and citizens, we are bound to perform. To these two forms are commonly given the not inappropriate names of spiritual and temporal jurisdiction, intimating that the former species has reference to the life of the soul, while the latter relates to matters of the present life, not only to food and clothing, but to the enacting of laws which require a man to live among his fellows purely honorably, and modestly. The former has its seat within the soul, the latter only regulates the external conduct. We may call the one the religious, the other the civil kingdom. Now, these two, as we have divided them, are always to be viewed apart from each other. Let us now return to human laws. If they are imposed for the purpose of forming a religious obligation, as if the observance of them was in itself necessary, we say that the restraint thus laid on the conscience is unlawful. Our consciences have not to do with men but with God only. Hence the common distinction between the earthly forum and the forum of conscience.
There is a twofold society, of which almost all men in the world are members, and from that twofold concernment they have to attain a twofold happiness; viz. That of this world and that of the other: and hence there arises these two following societies, viz. religious and civil.
In Roman Catholicism 
The Catholic Church has a similar doctrine called the doctrine of the two swords, in the bull Unam Sanctam, issued by Pope Boniface VIII. In this bull, Boniface teaches that there is only one Kingdom, the Church, and that the Church controls the spiritual sword, while the temporal sword is controlled by the State, although the temporal sword is hierarchically lower than the spiritual sword, allowing for Church influence in politics and society at large.
See also 
- Law and Gospel
- Render unto Caesar... for one of the passages this theology was derived from
- Christianity and politics
- Political catholicism
- Cf. Formula of Concord, article VI "The [Lutheran] Third Use of the Law.  Section 9
-  St. Louis edition of Luther's Werke, vol XI pp1726ff 1900
- Madison to Schaeffer, 1821
- John Locke, On the Difference between Civil and Ecclesiastical Power
- Lutheran Church of Australia: Commission on Social and Bioethical Questions THE TWO 'KINGDOMS' (2001)