The trinity is not a Biblical teaching. Both Peter and Stephen testified that they saw Jesus standing on the right hand of God. No verses in the Bible formally outline the trinity as doctrine and Jesus Christ prayed that the apostles would be one in the same way God and Jesus were one, demonstrating that His statement that He and His Father were one did not signify a trinity, but a unity of purpose, love, and doctrine.
The term “trinity” is not found in the Bible. Mormons prefer to use the term Godhead, which is used in the Bible. The Godhead consists of God the Father, His Son Jesus Christ, and the Holy Ghost. Each are entirely unified in every way except bodily. They are distinct and separate individuals with God at the head.
Mormons believe that God created our spirits. We lived with him for a time, a doctrine explained in more detail in the article on the Plan of Salvation. This makes Him very literally our Father in Heaven who loves us. Mormons believe God is perfect and that His plan is completely fair and loving to each of His children, regardless of the seemingly uneven circumstances of their births.
Mormons believe that Jesus Christ is the literal Son of God. He was born to a mortal mother, Mary. It is not true that Mormons believe there was an intimate relationship between them, nor do Mormons believe the Holy Ghost is Jesus’ Father. We do not know how the conception occurred beyond what the Bible tells us, but that information is unimportant. What is important is that Jesus Christ came to earth with a genetic heritage that was both divine and mortal. This made possible the atonement that altered eternity for all mankind.
Jesus Christ came to earth to atone for sins. This topic is covered further in Are Mormons Christian? We could not save ourselves because, as fully mortal beings, we are unable to live perfect lives. The plan of salvation requires perfection to satisfy justice, but allows for a Savior to satisfy justice. Only Jesus Christ could fulfill the role of Savior. Without His willingness to come to earth, suffer, atone for our sins, and die on the cross, we could not be saved and return to God someday.
The role of the Holy Ghost is to provide a path of communication between God and mankind. He is the only member of the Godhead who does not have a perfected body of flesh and bones (which is how Peter and Stephen were able to see them) but is instead a spirit. This allows Him to be many places and serving many people at the same time.
The Holy Ghost testifies of truth and is the way we can know God and Jesus Christ are real and to recognize eternal truths when they are presented to us. Mormons believe everyone is entitled to this help, but after baptism and confirmation, the Holy Ghost can stay with us at all times, as long as we are living worthy of His presence. For Mormons, baptism occurs at the age of eight, which is the age of accountability. Mormons spend a great deal of time preparing their children for this event so they can make an informed decision and be prepared to make and keep the covenants of baptism voluntarily.
The Godhead is God’s perfect system for carrying out the work of the church and for giving us every possible opportunity to choose God and Jesus Christ.