Ask a Latter-day Saint about the Trinity and the answer will usually begin by declining the word. The movement teaches that the Father, the Son and the Holy Ghost are three distinct beings, entirely united in will, purpose and love, and that this unity is what the New Testament means when it says they are one. It calls the arrangement the Godhead. The Nicene formula of one substance in three persons is rejected explicitly, and so is the accompanying claim that God is without body, parts or passions.
The positive teaching is unusually concrete. Doctrine and Covenants 130:22, recorded in 1843, states that the Father has a body of flesh and bone as tangible as a man's, that the Son likewise has one, and that the Holy Ghost is a personage of spirit, which is what allows him to dwell in a person. Joseph Smith's account of the First Vision, in which he described seeing the Father and the Son as two separate personages, is treated as the founding evidence for the distinction rather than as a devotional story.
Several other Latter-day Saint positions follow from this one and are difficult to state without it. Human beings are held to be the literal spirit offspring of God rather than creatures made from nothing, so the difference between divine and human nature is one of degree and exaltation rather than of kind. Prayer is addressed to the Father in the name of the Son, never to Jesus directly. And Latter-day Saint teaching about Heavenly Mother, which the church affirms while saying very little about her, has room to exist at all only because the Godhead is conceived in these terms.