If Latter-Day Saints (Mormons) don't believe in a Holy Trinity, then who do they believe is God? It's a pretty well-known fact that Latter-Day Saint members (aka Mormons) do not believe in the Holy Trinity, i.e., that God, Christ and the Holy Ghost are 'one'.  So what exactly do they believe in, when it comes to the nature of God?  

Along these lines, how can they understand scriptures that state Christ and God are one?
LDS understanding of the nature of the Father and the Son is drawn from the First Vision, Joseph Smith's first-hand account of an encounter with them as an answer to his prayer to learn the truth about God and religion:


  I saw a pillar of light exactly over my head, above the brightness of the sun, which descended gradually until it fell upon me. ... When the light rested upon me I saw two Personages, whose brightness and glory defy all description, standing above me in the air. One of them spake unto me, calling me by name and said, pointing to the other—This is My Beloved Son. Hear Him!


The Father and the Son reveal themselves as distinct individuals.  And if this is not in harmony with the traditional description passed down in various creeds,


  The Personage who addressed me said that all their creeds were an abomination in his sight; that ... “they draw near to me with their lips, but their hearts are far from me, they teach for doctrines the commandments of men, having a form of godliness, but they deny the power thereof.”


Joseph Smith later clarified that:


  The Father has a body of flesh and bones as tangible as man’s; the Son also; but the Holy Ghost has not a body of flesh and bones, but is a personage of Spirit. Were it not so, the Holy Ghost could not dwell in us.


LDS understanding of the Father and the Son being "one" is more in keeping with the literal meaning of the original text than of the interpretations that arose from it.  It kind of gets lost in translation, but it's a lot more clear in Greek that the Father and Christ are not one (the same being); they are one (united, the same in purpose.)  Further clarification can be found in John 17, where Jesus asks four separate times in his great intercessory prayer to the Father that his disciples may be one "even as we are."  To interpret this as a desire for the disciples to somehow merge into a composite being would be strange, to say the least.