From Anstey’s Doctrinal Definitions:
This term is not found in Scripture, but the truth that it conveys most certainly is. It refers to the Lord’s taking humanity into union with His Person and becoming a Man. This means that He had a human “spirit” (John 11:33; 13:21), a human “soul” (Matt. 26:38; John 12:27), and a human “body” (Heb. 10:5). Yet in doing this, He did not give up His divinity. Thus, He was fully a Man and fully God. This union of the divine and human natures is inscrutable to the mind of man (Matt. 11:27). We are not asked to understand it, but simply to accept and believe it. This incarnation occurred when Mary “conceived” of the Holy Spirit (Matt. 1:20) and He was “born” into the world (Luke 1:35). See John 1:14; Romans 1:3; Philippians 2:5-8; 1 Timothy 3:16a.
The Lord did not simply take a human body as a shell and have His divine Spirit dwell in it—the error of Apollinarianism. There was an actual union between all that there is in Him as a divine Person with all that constitutes a human being. Becoming a Man was an incredible condescension, for this union of the divine and human natures is something that will be forever. The Lord lived and moved and had His being in this world as a Man, and He laid down His life in death as a Man, and thus He passed into the intermediate state (His soul and spirit being temporarily separated from His body), but even in death He was still a Man (Acts 2:27). In resurrection, He took His body again and ascended to the right hand of God in a glorified state (1 Tim. 3:16b)—and He will remain a Man for all eternity!
When Scripture speaks of Christ’s humanity, it carefully guards against the notion that He took part in the fallen sin-nature which we have. For instance, Hebrews 2:14 says, “As the children are partakers of flesh and blood, He also Himself likewise took part in the same.” This means that “as” humans take part in manhood (which involves three parts—a human spirit, a human soul, and a human body), Christ also took part “in the same.” In other words, He became a real Man. However, in speaking of Christ’s humanity, the Spirit uses a different word from what He uses to describe our humanity. The "children," the writer says, are “partakers” of flesh and blood. The word in the Greek translated “partakers” (koinoneo) refers to a common equal sharing in humanity. This is true of all men, for we all fully partake in humanity—right down to partaking in the sin-nature. However, when the verse goes on to speak of Christ becoming a Man, the writer uses another Greek word. He says that that the Lord “took part” (metecho) in the same. This word indicates a sharing in something without specifying what degree the sharing went to, and thus points to the fact that when Christ became a Man, His humanity did not go so far as to share in fallen humanity.
Also, Hebrews 4:15 states that the Lord’s humanity was “sin apart.” “Sin” (singular) is often used in the New Testament epistles to indicate the old, fallen sin-nature. This verse, therefore, makes it clear that the Lord didn’t have a fallen human nature, as all the descendants of fallen Adam do (Psa. 51:5).
The incarnation, therefore, is different from the theophanies in the Old Testament. These were occasions when Christ took a human form and interacted with men for certain purposes. Oftentimes Scripture uses the title “Angel” to indicate this (Gen. 16:7; 18:1-33; 32:24-32; Ex. 3:2; 4:24; 14:19; Josh. 5:13-15; Judg. 6:11-24; 13:3-5; 9-21; 1 Chron. 21:18-30, etc.). On those occasions, the Lord appeared in a human form, but it was not the union of the two natures (divine and human) involved in His incarnation.