What does the LDS/Mormon church teach about the relationship between Jesus and Lucifer? Are they beings on the same level or of the same origin? What are their natures? What texts are used to teach specifically about the relationship between them?
The LDS theological view is that God, the Eternal Father, created all spirits, including Jesus the Firstborn, Lucifer, and all the rest of us, as His spirit children. Further, God is a separate being from Jesus and the Holy Ghost, who are all individual personages.
These three beings constitute the godhead, which is different from the usual Trinity belief.
These teachings are in harmony with at least the following scriptures:
For context: there was a council in heaven regarding God's plan for all His spirit children. God's plan was presented, and a Savior was needed to fulfill it. Jesus was Chosen by the Father to fulfill that role.
Satan opposed, declaring his plan, which was contrary to the will of the Father. War ensued between "the dragon" (Satan) and the Christ's followers, which war is still ongoing today, here in mortality. It's a sibling rivalry, if you will, which Jesus and His disciples will win in the end.
The LDS church teaches that Christ, Lucifer, and all beings that live know, have lived and will live here are God's spirit children. That makes Christ and Lucifer brothers, and all of us their siblings.
Before we came to earth God asked one of His children to play the role of a savior. That person would have to carry out the atonement so that both justice and mercy could be satisfied.
There were two volunteers. Jesus Christ was willing to obey God's plan. This plan allows the rest of us to choose whether or not we would follow God's commandments and give the glory to God.
Lucifer wanted to force us to obey the commandments and he wanted the glory for himself. One third of God's children wanted it Satan's way and a war broke out.
Lucifer and his followers were cast out and not allowed to gain a body and live a life on earth.
So the LDS church teaches that Lucifer is our spiritual brother and Christ's as well. That doesn't mean that he has our best interests in mind. He's jealous of our physical bodies and the experiences we get to have and so he and his followers try to get us to abuse the bodies we have.