If Latter-Day Saints (Mormons) don't believe in a Holy Trinity, then who do they believe is God?

Mormons believe in "one God", however, they do not believe in an "ontological oneness." In other words, God the Father, God the Son, and God the Holy Ghost are three separate personages, but they are "One God." An analogy I have always been partial too is if 3 men were walking down the street side by side, and you were following them, who are you following? This "oneness" of purpose, unity and direction is what Christ alludes to in His intercessory prayer in John 17, when He petitions the Father to bless his disciples "that they may be one, as you and I are one".

Compare also the "one" use in the Shema of the Torah (Deut 6:4 -The Lord God is one) to the "one" used in Genesis to describe a husband and wife (therefore shall a man cleave unto his wife, and they shall become one flesh.) The Hebrew word translated as "one" in English is the same word in Hebrew in both instances.

The official doctrine of the Church of Jesus Christ of Latter-day Saints is that there are 3 separate and distinct Gods, that comprise "the Godhead". It teaches in a literal "plurality of Gods" (as founder Joseph Smith phrased it and used in official sources). [1]

The Church and its members do sometimes say they are "one God", but this is figurative, not literal. The "one God" phrase by the LDS Church means they are all in complete unity of purpose, etc. This is not the historic Trinity doctrine.

Here are official sources from the Church of Jesus Christ of Latter-day Saints, the LDS Church or Mormon Church (the latter are nicknames), that I have researched personally:

“… the head God brought forth the Gods in the grand council. “… The head God called together the Gods and sat in grand council to bring forth the world. The grand councilors sat at the head in yonder heavens and contemplated the creation of the worlds which were created at the time.” (Teachings, pp. 348–49.) The Abraham account of the Creation reflects this idea of the plurality of Gods (see Abraham 4).” [2]

"Heavenly Father, Jesus Christ, and the Holy Ghost are all Gods." [3]

"I have always declared God to be a distinct personage, Jesus Christ a separate and distinct personage from God the Father, and that the Holy Ghost was a distinct personage and a Spirit: and these three constitute three distinct personages and THREE Gods" [4]

→ LDS scripture: * Abraham chapter 4 summary: “The Gods plan the creation of the earth and all life thereon—Their plans for the six days of creation are set forth.” [5]

Abraham 4:1,5,26: “the Gods, organized and formed the heavens and the earth. …. And the Gods called the light Day, and the darkness they called Night. …. And the Gods took counsel among themselves” [6]

Abraham chapters 4 & 5 repeatedly use “the Gods” for those beings engaged in creation of heaven and earth.

→ Other official LDS Church sources:

Founding prophet Joseph Smith taught concerning the nature of the Father, the Son, and the Holy Ghost:

“these three constitute three distinct personages and three Gods.”

These 3 Gods are the Godhead who are in perfect agreement, as taught in a subtitle in an LDS Church manual:

“The Godhead is in perfect unity, and God the Father presides.” [8]

Joseph Smith taught in the 1844 Elder King Follett funeral sermon the official doctrine that God the Father “dwelt [lived] on an earth” [9]

““God Himself was once as we are now, and is an exalted man, and sits enthroned in yonder heavens!” [10]

Our potential is to become Gods/gods as he has done, both men and women:

From “The Presidents of the Church: Lorenzo Snow”, an official manual in Chapter 5 “The Grand Destiny of the Faithful”, the Church quotes LDS President Snow and also explains this “doctrine” of the Church:

“I [Lorenzo Snow] formed the following couplet which expresses the revelation, as it was shown me. … ““As man now is, God once was: ““As God now is, man may be.” “Feeling that he had received “a sacred communication” that he should guard carefully, Lorenzo Snow did not teach the doctrine publicly until he knew that the Prophet Joseph Smith had taught it. Once he knew the doctrine was public knowledge, he testified of it frequently.” [11]

The official magazine, the “Ensign”, February 18, 1982 affirms that President Snow’s couplet and doctrine was indeed “true” and a “revelation from God”:

“President Snow’s son LeRoi later told that the Prophet Joseph Smith confirmed the validity of the revelation Elder Snow had received: “Soon after his return from England, in January, 1843, Lorenzo Snow related to the Prophet Joseph Smith his experience in Elder Sherwood’s home. This was in a confidential interview in Nauvoo. The Prophet’s reply was: ‘Brother Snow, that is a true gospel doctrine, and it is a revelation from God to you.’” (LeRoi C. Snow, Improvement Era, June 1919, p. 656.) …. ““This is a doctrine which delighted President Snow, as it does all of us [Latter-day Saints/’Mormons’]. Early in his ministry he received by direct, personal revelation the knowledge that (in the Prophet Joseph Smith’s language), ‘God himself was once as we are now, and is an exalted man, and sits enthroned in yonder heavens,’ and that men ‘have got to learn how to be Gods … the same as all Gods have done before.’ … “It is clear that the teaching of President Lorenzo Snow is both acceptable and accepted doctrine in the Church today.” [12]

→ FOOTNOTES:

[1] - “Ensign”, February 1994, “I Have a Question”. [1] - “Old Testament Manual Genesis 22 Samuel”, “Enrichment Section A).

[2] - “Old Testament Student Manual Genesis-2 Samuel: Religion 301” (2003, LDS “Church Educational System”.

[3] - "New Testament Stories", "People to Know", p 161 (for kids).

[4] - Joseph Smith, quoted in the Ensign, March 2008, p 68, Teachings of Presidents of the Church: Joseph Smith [2007], pp 41-42, see also: Aaronic Priesthood Manual 3, Lesson 17: "The Holy Ghost", p 63. (See also History of the Church, 6:474, Teachings of the Prophet Joseph Smith, p. 372).

[5] – LDS scripture: “Book of Abraham” Chapter 4 Summary.

[6] – LDS scripture: “Book of Abraham” 4:1,5,26.

[7] - “Teachings of the Presidents of the Church: Joseph Smith”, chapter 2.

[8] - Chapter 2 subheading in “Teachings of the Presidents of the Church: Joseph Smith”, chapter 2.

[9] – Joseph Smith, King Follett Discourse 1844, LDS Church manual, “Gospel Principles”, 2009, p. 279.

[10] - “Teachings of the Presidents of the Church: Joseph Smith”, chapter 2.

[11] - “Teachings of the Presidents of the Church: Lorenzo Snow”, Chapter 5: “The Grand Destiny of the Faithful”.

[12] – Gerald N. Lund, Teacher Support Consultant for the Church Education System, “Ensign” February 1982, “I Have a Question”: “Is President Lorenzo Snow’s oft-repeated statement—“As man now is, God once was; as God now is, man may be”—accepted as official doctrine by the Church?”.

Disclaimer:I am not a Mormon, so this may be inaccurate.

The first place to read is what Mormons say about this themselves. The Wikipedia article is a good place to start and has references. In essence they believe that the Father Son and Holy Spirit are separate "personages", and together form "The Godhead", which is a council of the three personages. The scriptures that refer to the distinct personages "being one" are taken to mean "of one mind".

This is based on Joseph Smith's "Lectures on Faith". See the text