The 2002 LDS booklet titled Preparing to Enter the Holy Temple states:
If we would understand both the history and the doctrine of temple work we must understand what the sealing power is. We must envision, at least to a degree, why the keys of authority to employ the sealing power are crucial.
Nearly nine hundred years before Christ, the prophet Elijah appeared in the court of the king of Israel. He carried with him a sacred authority: the power to seal.
Elijah worked out his ministry, ordained and anointed Elisha to succeed him, and then—and this is important—he did not die. Like Moses before him, he was translated.
After that, his name appears only once in the Old Testament, in the next to the last verse of the last chapter of the Old Testament. It is here that Malachi prophesies that Elijah would return and that he would “turn the heart of the fathers to the children, and the heart of the children to their fathers,” lest the whole earth be smitten with a curse. (See Malachi 4:5–6.)
Malachi 4:5-6 says,
See, I will send the prophet Elijah to you before that great and dreadful day of the Lord comes. He will turn the hearts of the parents to their children, and the hearts of the children to their parents; or else I will come and strike the land with total destruction.
According to the 2009 church manual Gospel Principles, “Elijah would restore the sealing powers so families could be sealed together. He would also inspire people to be concerned about their ancestors and descendants.” BYU professor Charles R. Harrell explains, “Latter-day Saints view this prophecy as having been fulfilled with the coming of Elijah to the Kirtland Temple on April 3, 1836 (D&C 110:13-16).” (This is My Doctrine, p. 74). However, he says this is not what Malachi was talking about, saying, “Malachi’s prophecy, as it appears in the Old Testament, gives no indication that Elijah would restore priesthood keys or reveals sealing ordinances either for the living or the dead” Ibid).
Instead, according to the New Testament, this person is John the Baptist (Matthew 11:14; 17:10; Mark 9:11-13; Luke 1:17). Jesus said in Matthew 11:
11 Truly I tell you, among those born of women there has not risen anyone greater than John the Baptist; yet whoever is least in the kingdom of heaven is greater than he. 12 From the days of John the Baptist until now, the kingdom of heaven has been subjected to violence, and violent people have been raiding it. 13 For all the Prophets and the Law prophesied until John.14 And if you are willing to accept it, he is the Elijah who was to come. 15 Whoever has ears, let them hear.
In case the reader didn’t understand the meaning behind this teaching, Jesus provided more information in Matthew 17:10-13. It says,
10 The disciples asked him, “Why then do the teachers of the law say that Elijah must come first?” 11 Jesus replied, “To be sure, Elijah comes and will restore all things. 12 But I tell you, Elijah has already come, and they did not recognize him, but have done to him everything they wished. In the same way the Son of Man is going to suffer at their hands.” 13 Then the disciples understood that he was talking to them about John the Baptist.
Harrell agrees, adding,
John’s coming in the spirit of Elijah, therefore, was for the purpose of turning the affections of parents to their children and, in general, turning the disobedient to God. There is no mention of John exercising what Mormons traditionally understand as sealing powers, but then, according to LDS doctrine, he would have been unable to because he held only the Aaronic Priesthood (This is My Doctrine, pp. 74-75).
Harrell then explains that this passage was never understood the way it is now interpreted by Mormons:
Significantly, in all versions of Malachi’s prophecy revealed to Joseph Smith prior to 1838, there is no change in the wording from the the KJV, nor is there any indication that Elijah would restore any sealing authority. This includes passages in the Book of Mormon (3 Ne. 25:5-6), the Doctrine and Covenants (D&C 27:9, 110:13-16), and even JST Malachi 4:4-5. On July 2, 1833 Joseph dictated the word “correct,” which his scribe wrote at the head of the book of Malachi indicating his approval of the text as it stood. Even the account of Elijah’s return to the Kirtland Temple in April 1836 states only that the keys he restored were intended “to turn the hearts of the fathers to the children, and the children to the fathers” (D&C 110:14). There is no mention of a conferral of “sealing” powers.”
It wasn’t until 1838 when Smith changed his view. Then, in September 1842. “the first explicit association was made between the coming of Elijah and salvation for the dead.”