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Child of God
Many friends of other faiths may connect the Mormon Primary children's hymn, “I Am a Child of God,”  with Mormons, because members have sung it widely since 1957. This concept—that each person here on earth is a child of a Heavenly Father—the Most High God—is central to the Mormon religion (Psalms 82:6 “I have said, Ye are gods; and all of you are children of the most High.”). To members of The Church of Jesus Christ of Latter-day Saints, their Father is not an anonymous being who simply wields power over their lives but a real individual whom they have always known and loved. Their relationship with him is very personal.
This hymn relates many things about this relationship. First, Mormons believe that God the Eternal Father is the one who sends babies to earth to various circumstances and types of families. To some non-members, this could be viewed as an indictment against God because many children are sent to bad situations. Mormons, on the other hand, trust in his wisdom, love, and mercy. They believe that, as their parent, he knows his children, what they need to learn, and what they are capable of enduring and overcoming, also alluded to in the song. However, in addition to this, the hymn also stresses the responsibility that parents—and others—have in teaching these children how to return to their Heavenly Father. Mormons believe that those who obey the commandments given to them for their welfare and guidance will receive great blessings when their lives are over.
Spirit Children/Sons of God
Mormons sometimes consider Sons of God as synonymous with Children of God. Job 38:7 says, “When the morning stars sang together, all the sons of God shouted for joy”... “when the foundations of the earth” (Job 38:4) were laid. Mormons believe that those sons of God are his spirit offspring who rejoiced for the opportunity to receive physical bodies like their Father, and experience a mortal probation designed to help them grow to spiritual maturity as he did. Mormons believe that, as filled with grief and pain—and as brief--as life can be, this opportunity is desirable enough to warrant it. (See Plan of Salvation.)
Earthly Children/Sons of God
Earthly sons of God, more specifically defined, are the ones who accept Christ, obey his commandments, progress and endure till they are spiritually begotten of Christ. Latter-day Saints believe that individuals choose who their spiritual father here on earth will be: Christ or Lucifer, simply through whom they choose to obey and follow.
Power of Children/Sons of God
Mormons believe in the power to become sons of God that is given to them when they convert to the Church of Jesus Christ of Latter-day Saints. This conversion must come even to those who are born into the Church, because it doesn't automatically come with membership alone (Mormon Doctrine, p. 745). However, the apostle Bruce R. McConkie says, “admission into the Church opens the door to such high status [to be sons of God], if it is followed by continued faith and devotion (Romans 8---:14-18; Galatians 3:26-29; 4:1-7).” He also points out that as “members of his family” they “are joint-heirs with Christ, inheriting with him the fullness of the Father (Doctrine and Covenants 93:17-23).”
McConkie emphasizes that, to be sons of God and joint-heirs with Christ, men must obtain the higher Melchezidek priesthood (the same priesthood Christ bears), magnify their calling in it, enter into the new and everlasting covenant of marriage, and be obedient in all things. He says that “those who become the sons of God in this life (1 John 3:1-3) are the ones who by enduring in continued righteousness will be gods in eternity" (Doctrine and Covenants 76:58). So, while Mormons believe that all men are sons of God in the pre-mortal existence because there they were "born" as spirit children to him, whether they remain sons of God in the hereafter will depend upon their fulfilling His requirements here on earth.