The "Hypostasis" describes that Jesus Christ had not one,  but two natures as one single person:  A human nature, and a divine nature.  It follows also that He had two wills - a divine will and a human will.  Those who insisted Our Lord had only one will were heretics called Monophysites. 

What does it mean to have two natures?  Well, in Christ's divine nature, He was never separated from the Trinity, of which He is the Second person.  Imagine, if you can, Our Lord having continual access to the power He possesses as God, while condescending to also possess a fragile human estate subject to bodily pain, corruption, temptation, emotions, and fears.  

Christ, in effect, chose to take on the flesh: to grow from an embryo to an adult, traversing every state of human development along the way.  His human and experiential knowledge grew despite His own omniscience.  He learned the craft of carpentry;  He read and learned the Hebrew Scriptures;  He discovered the world around Himself as a child, and only when necessary to the divine plan, exposed His divinity.  Thus He taught the elders in the Temple as a sign of not only His human maturity, but the undertaking He must one day fulfill - to be the Temple Himself, to give His greatest teaching from the instrument of His death.  He suffered all the pain of human suffering, while not "grasping at equality with God".  His sacrifice was total, consuming both his humanity and His Godhead.  Indeed He suffered as the God-Man, and died as the God-Man.  God, it can be said without scandal, died on the Cross for our sins. 

"Have among yourselves the same attitude that is also yours in Christ Jesus,
Who, though he was in the form of God,
did not regard equality with God something to be grasped.
Rather, he emptied himself,
taking the form of a slave,
coming in human likeness;
and found human in appearance..."
Philippians 2:5-7

464 The unique and altogether singular event of the Incarnation of the Son of God does not mean that Jesus Christ is part God and part man, nor does it imply that he is the result of a confused mixture of the divine and the human. He became truly man while remaining truly God. Jesus Christ is true God and true man.

465 The first heresies denied not so much Christ's divinity as his true humanity (Gnostic Docetism). From apostolic times the Christian faith has insisted on the true incarnation of God's Son "come in the flesh".  But already in the third century, the Church in a council at Antioch had to affirm against Paul of Samosata that Jesus Christ is Son of God by nature and not by adoption. The first ecumenical council of Nicaea in 325 confessed in its Creed that the Son of God is "begotten, not made, of the same substance (homoousios) as the Father", and condemned Arius, who had affirmed that the Son of God "came to be from things that were not" and that he was "from another substance" than that of the Father.

466 The Nestorian heresy regarded Christ as a human person joined to the divine person of God's Son. Opposing this heresy, St. Cyril of Alexandria and the third ecumenical council, at Ephesus in 431, confessed "that the Word, uniting to himself in his person the flesh animated by a rational soul, became man." Christ's humanity has no other subject than the divine person of the Son of God, who assumed it and made it his own, from his conception. For this reason the Council of Ephesus proclaimed in 431 that Mary truly became the Mother of God by the human conception of the Son of God in her womb: "Mother of God, not that the nature of the Word or his divinity received the beginning of its existence from the holy Virgin, but that, since the holy body, animated by a rational soul, which the Word of God united to himself according to the hypostasis, was born from her, the Word is said to be born according to the flesh."