In 451 A.D. the Council of Chalcedon was convened by Emperor Marcion at the request of Pope Leo the Great. The settlement is considered to be the high-water mark of the early church’s christological speculation. It was formulated against the backdrop of nearly four centuries of controversy regarding the person of Christ. For a statement of the Chalcedonian definition, see below.
I thought I would contribute something that has helped me understand the Chalcedonian definition regarding the Incarnation of the Son of God. Comments made to me of late speak of it being very difficult to understand, and I would beg to dissent. I admit that the antiquated language is difficult. The obscure terms are difficult. Also it is difficult in that it says what it wants to say in a long, drawn-out way – which is often the way of philosophical treatises that desire precision. But the idea itself seems to me be easily grasped.
What it was that helped me was the following. 
The settlement is a ringing endorsement of dyophysite [two-nature] Christology. Christ is declared to exist in two natures, whose distinction remains real even in their union with Christ. . . At the same time, however, in agreement with monopysite [one-nature] Christology, the settlement insists on there being only one person, one Son, in Christ. . . . Person and hypostasis are taken as having the same referent, so that the Incarnation becomes a sort of mirror image of the Trinity. Just as in the Trinity there are multiple persons in one nature, so in Christ there are multiple natures in one person. The famous series of the four adjectives asynchytos, atreptos, adiairetos, achoristos (without confusion, without change, without division, without separation) serves as a reminder that the two natures of Christ must be kept distinct and that the the unity of his person must not be compromised. . . . As a result of Chalcedon, it has become an imperative of orthodox Christology that we must “neither confuse the natures nor divide the person” of Christ.
The Chalcedonian formula itself does not tell us how to do this. It does not seek to explain the Incarnation but sets up, as it were, channel markers for legitimate Christological speculation; any theory of Christ’s person must be one in which the distinctness of both natures is preserved and both meet in one person, one Son, in Christ. It admittedly fulfilled the purpose for which it was drawn up; namely, to exclude two possible but unacceptable explanations of the Incarnation and to provide a convenient summary of essential facts that must be borne in mind by all those who attempt to penetrate further into the mystery. 
The question that Chalcedon is answering then is not, “How is it that Jesus can be God and human at the same time?” which, I admit, is difficult. Note the illustration of channel markers. In effect this says that there can be a wide variety of how to answer this question, as long as one rows their boat of speculation between the two borders marked out for them. The question that Chalcedon answers then is rather, “What are the boundaries to acceptable speculation regarding the person of Christ?”
Alternatively, one could say the question was, “How is it not a logical contradiction that Jesus can be fully God and fully human at the same time?” And the Chalcedonian definition avoids any logical contradiction in that two natures are attached in some way (perhaps we may never know exactly how) to one person. Not two people in one person, nor two natures in one nature, which would both be logically contradictory, but two natures in one person. And even if you don’t understand what a nature is or a person is, as Roger E. Olson explains, it is two whats and one who.  Thus the doctrine of the Incarnation can be rationally affirmed.
1. From C.R.T.A.: The Centre of Reformed Theology and Apologetics.
Therefore, following the holy fathers, we all with one accord teach men to acknowledge one and the same Son, our Lord Jesus Christ, at once complete in Godhead and complete in manhood, truly God and truly man, consisting also of a reasonable soul and body; of one substance with the Father as regards his Godhead, and at the same time of one substance with us as regards his manhood; like us in all respects, apart from sin; as regards his Godhead, begotten of the Father before the ages, but yet as regards his manhood begotten, for us men and for our salvation, of Mary the Virgin, the God-bearer; one and the same Christ, Son, Lord, Only-begotten, recognized in two natures, without confusion, without change, without division, without separation; the distinction of natures being in no way annulled by the union, but rather the characteristics of each nature being preserved and coming together to form one person and subsistence, not as parted or separated into two persons, but one and the same Son and Only-begotten God the Word, Lord Jesus Christ; even as the prophets from earliest times spoke of him, and our Lord Jesus Christ himself taught us, and the creed of the fathers has handed down to us.
2. The whole chapter called Christian Doctrines (II): The Incarnation is long, but thoroughly worth reading in my opinion. (see footnote 3).
3. J.P. Moreland and William Lane Craig, Philosophical Foundations for a Christian Worldview, IVP, 2003, p. 601.
4. Roger E. Olson, Mosaic of Christian Belief: Twenty Centuries of Unity and Diversity, IVP Apollos, 2002, p. 227.