After Moses, came Elias. We know not who he was in mortality. There were many prophets who bore that name and title. One was Noah. Apparently this Elias lived in the day of Abraham, and may even have been Abraham himself. In any event he “committed the dispensation of the gospel of Abraham” (D&C 110:12)—not, be it noted, the gospel of Christ, for that had already been received, but the gospel of Abraham, meaning the great commission which God gave Abraham in his day. That commission dealt with families, those of Abraham and his seed, who were and are promised continuance “in the world and out of the world … as innumerable as the stars; or, if ye were to count the sand upon the seashore ye could not number them” (D&C 132:30).
As Joseph Smith records it, what Elias actually said to him and Oliver Cowdery was that “in us and our seed all generations after us should be blessed” (D&C 110:12). And so, the Lord be praised, the marriage discipline of Abraham was restored; it is the system that enables a family unit to continue in eternity; it is the system out of which eternal life grows. The laws and conditions pertaining to this holy order of matrimony in the new and everlasting covenant constitute “the promises made to the fathers,” which, as Moroni said (quoting Malachi), were to be planted “in the hearts of the children” (D&C 2:2). ("This Final Glorious Gospel Dispensation," Ensign, April 1980, 21)
The man Elias brings back "the gospel of Abraham," the great Abrahamic covenant whereby the faithful receive promises of eternal increase, promises that through celestial marriage their eternal posterity shall be as numerous as the sands upon the seashore or as the stars in heaven for multitude. Elias gives the promise—received of old by Abraham, Isaac, and Jacob—that in modern men and in their seed all generations shall be blessed. And we are now offering the blessings of Abraham, Isaac, and Jacob to all who will receive them. ("The Keys of the Kingdom," Ensign May, 1983, p. 21-23)
[T]he Lord sent Elias and he sent Elijah. And when Elias came, he brought the gospel of Abraham, the divine commission that God gave Abraham, the marriage discipline that God gave Abraham. Elias restored celestial marriage, and Elijah came and brought the sealing power so the ordinance would be binding on earth and sealed in heaven; and it takes the ministry of both of them to accomplish the purposes of the Lord. ("The Promises Made to the Fathers," From Studies in Scripture Vol 3: Genesis to 2 Samuel,
ed. Kent P. Jackson and Robert L. Millet [Salt Lake City, UT: Randall Book Co., 1985], p. 61)
There are three definitions of Elias. Elias equals Elijah in those cases where it's talking about the latter. Elias equals Elias in those cases where it's talking about a particular man, a prophet—we know very little about him, but he existed. Third, Elias can mean "a messenger."...
[In the Kirtland Temple] Elias—the Elias, not a messenger but the prophet Elias—appeared and committed the keys of the dispensation of the gospel of Abraham. (The Things of the Soul, p. 208, 211)