What does it mean when Christians say that God is a person? (Representative example here.) Does this mean that Christians believe that God is human?
My experience is with Judaism (in Judaism, God is not referred to as a person).
If we are strict with our terminology, God is not a person. God is an infinitely-perfect being. But God is a personal being:
When we say that God is a personal being we mean that He is intelligent and free and distinct from the created universe. Personality as such expresses perfection, and if human personality as such connotes imperfection, it must be remembered that, as in the case of similar predicates, this connotation is excluded when we attribute personality to God. It is principally by way of opposition to Pantheism that Divine personality is emphasized by the Theistic philosopher. (source)
Your concern, however, is valid:
Yet sometimes men are led by a natural tendency to think and speak of God as if He were a magnified creature — more especially a magnified man — and this is known as anthropomorphism. Thus God is said to see or hear, as if He had physical organs, or to be angry or sorry, as if subject to human passions: and this perfectly legitimate and more or less unavoidable use of metaphor is often quite unfairly alleged to prove that the strictly Infinite is unthinkable and unknowable, and that it is really a finite anthropomorphic God that men worship.
When we refer to God as a person, we mean that He is personal, not that He is a magnified human.
When speaking of God, we usually try to reserve the term "Person" to one of the three Divine Persons of the Trinity. There is one God, who is a Trinity of three Divine Persons: the Father, the Son, and the Holy Spirit. The Son is Jesus Christ, who is true God and true man. When speaking of Jesus being a person, we say that He is a Divine Person, not a human Person, and we say that He is God, in perfect unity with the Father and the Holy Sprit.
That God is referred to as a Person is, first, in contrast to the idea of some that the universe was created by some impersonal force--a force that has no personal being, but just some kind of cosmic energy. Such a creative force would have created something greater than itself--beings with Personhood--so this is not reasonable.
The idea that God is a Person refers to the how He Himself describes Himself in His Word as well. God feels, loves, and relates to His creation. As A.W. Tozer wrote:
God is a Person, and in the deep of His mighty nature He thinks, wills, enjoys, feels, loves, desires and suffers as any other person may.
If God were not a Person, He could not love, as that is something only a Person can do. The doctrine of the Trinity is interesting in this aspect, because God reveals one of His attributes is love. However, if it is an attribute (and not a behavior) of God, it has to exist apart from anything else. As the Father loves the Son, the Son loves the Father; the Father loves the Spirit, etc. Thus, only as a Trinity can God be love from all eternity.
So, what is meant is that God is the source for the personal attributes that we as His creatures share in--that of loving, feeling, knowing, etc. In this we bear the image of God.
Scriptural references to the fact that we are children of God are ubiquitous (see Romans 8:16-17, Acts 17:28, 2 Corinthians 6:18, Galatians 3:26, and many other passages). The statement that man is created in the image of God (Genesis 1:26-27) is entirely literal. Moses saw the back of God, and talked with Him face to face; God has hands, eyes, feet and so on. The Savior testified with clarity while in His mortal body that "He that hath seen me hath seen the Father" (John 14:7,9).
Modern prophets including Joseph Smith and Ezra Taft Benson taught:
God himself was once as we are now, and is an exalted man, and sits enthroned in yonder heavens! That is the great secret. If the veil were rent today, and the great God who holds this world in its orbit, and who upholds all worlds and all things by his power, was to make himself visible,—I say, if you were to see him today, you would see him like a man in form—like yourselves in all the person, image, and very form as a man; for Adam was created in the very fashion, image and likeness of God. King Follett Sermon
When the Savior shall appear we shall see him as he is. We shall see that he is a man like ourselves. And that same sociality which exists among us here will exist among us there, only it will be coupled with eternal glory, which glory we do not now enjoy. enter link description here
When the Resurrected Lord appeared to His disciples, He said:
Behold My hands and My feet, that it is I Myself. Handle Me and see, for a spirit hath not flesh and bones, as ye see Me to have (Luke 24:39).
The notion that His body evaporated or dissolved and He turned into some sort of platonic, unknowable and disembodied thing is a patently false tradition of anti-Christians. He still lives in that same body today, and greatly resembles His Father as He said.
If we believe that He is without a body today then we are effectively perpetuating the lie told by the Roman officers to the soldiers, that Jesus is not risen, but that His body was merely stolen away by His disciples. But He is risen and lives forevermore. He does indeed have a face, a mind, and a complete and perfect body, lacking nothing. His presence is just as physical and real as that of any embodied person now on the Earth.