What does the LDS Church teach about the pre-existence of each person prior to conception?

This doctrine teaches, in short, that we literally existed prior to birth into mortality on this earth, and were involved in the salvation plan.

See the below excerpts from the Encyclopedia of Mormonism...

Pre-Existence (Pre-Earthly Existence)

The term "pre-existence," or more accurately, "premortal existence," refers to a period of individual conscious and accountable life before birth into mortality on this earth. It is Latter-day Saint doctrine that living things existed as individual spirit beings and possessed varying degrees of intelligence in an active, conscious spirit state before mortal birth and that the spirit continues to live and function in the mortal body. The revelations teach that premortal spirit bodies have general resemblance to their physical counterparts.

Council In Heaven

The Council in Heaven, sometimes called the Grand Council, refers to a meeting of God the Father with his spirit sons and daughters to discuss the terms and conditions by which these spirits could come to earth as physical beings. The terms "Council in Heaven" and "Grand Council" do not appear in the scriptures, but are used by the Prophet Joseph Smith in referring to these premortal activities, allusions to which are found in many scriptures (Job 38:4-7; Jer. 1:5; Rev. 12:3-7; Alma 13:3-9; D&C 29:36-38; 76:25-29; Moses 4:1-4; Abr. 3:23-28; cf. Tpjs, pp. 348-49, 357, 365; T&S 4 [Feb. 1, 1843]:82).

One purpose of the heavenly council was to allow the spirits the opportunity to accept or reject the Father's Plan of Salvation, which proposed that an earth be created whereon his spirit children could dwell, each in a physical body. Such a life would serve as a probationary state "to see if they [would] do all things whatsoever the Lord their God shall command them" (Abr. 3:25). The spirits of all mankind were free to accept or reject the Father's plan but they were also responsible for their choice. The Creation, the Fall, mortality, the Atonement, the Resurrection, and the Final Judgment were contemplated and explained in the council (TPJS, p. 220, 348-50; MD, pp. 163-64; see also First Estate). The plan anticipated mistakes from inexperience and sin and provided remedies. Many spirits were foreordained to specific roles and missions during their mortal experience, conditional upon their willingness and faithfulness in the premortal sphere and their promised continued faithfulness upon the earth. The Prophet Joseph Smith explained, "Every man who has a calling to minister to the inhabitants of the world was ordained to that very purpose in the Grand Council of heaven before this world was. I suppose I was ordained to this very office in that Grand Council" (TPJS, p. 365; cf. 1 Pet. 1:20; Jer. 1:5; Abr. 3:22-23).

Although spoken of as a single council, there may have been multiple meetings where the gospel was taught and appointments were made. Jesus and the prophets were foreordained in the council. A redeemer was to perform a twofold mission in redeeming mankind from the physical and spiritual deaths brought about by the Fall of Adam and also in providing redemption, upon repentance, for sins committed by individuals. At a certain point in the council, the Father asked, "Whom shall I send [as the Redeemer]?" Jesus Christ, known then as the great I AM and as Jehovah, answered, "Here am I, send me," and agreed to follow the Father's plan (Moses 4:1-4; Abr. 3:27). As a counter-measure, Lucifer offered himself and an amendment to the Father's plan of saving mankind that would not respect their agency. The substitute proposal was also designed to exalt Lucifer above the throne of God. The Father's response was, "I will send the first" (meaning Jehovah). Lucifer rebelled and became Satan, or "the devil." A division developed among the spirits, and no spirits were neutral (DS 1:65-66). There was war in heaven (Rev. 12:7-8), and the third of the hosts who followed Lucifer were cast out (Rev. 12:4; D&C 29:36). These rebellious spirits, along with Lucifer, were thrust down to the earth without physical bodies (Rev. 12:9; cf. Isa. 14:12-17). The Prophet Joseph Smith explained: "The contention in heaven was-Jesus said there would be certain souls that would not be saved; and the devil said he could save them all, and laid his plans before the grand council, who gave their vote in favor of Jesus Christ. So the devil rose up in rebellion against God, and was cast down, with all who put up their heads for him" (TPJS, p. 357). Heavenly Father and the faithful spirits in heaven wept over them (D&C 76:25-29). Satan and his followers are still at war with those spirits who have been born into mortality (Rev. 12:9; cf. "War in Heaven," p. 788).