This is certainly a strange question to ask. I wouldn't even include a reference to this idea except that 2 different people have asked me this same question over the past few months. I can only imagine that somewhere out there, someone got their hands on what they think is a juicy little piece of 'hidden Mormon doctrine' and is spreading it around like they're revealing some great mystery about what Mormon's 'really believe'.
I suppose some people imagine that when we speak about building strong families, building good moral characters, keeping the Lord's commandments and worshipping Jesus Christ - it's all just some kind of facade. Why do some have to keep trying to search for what we 'really believe'? Why don't they just go to a sacrament meeting and find out? Why are they not satisfied when we tell them what we believe? It's not an accurate representation of our doctrine, and it certainly is not something I've ever heard at church - and I've been an active member of the Mormon church all my life.
This may reflect a statement made by Brigham Young or some other early church leader in a larger address. But this is an obscure statement, which would have been made to an audience who already was very familiar with the larger doctrinal context which I explain below. When viewed within it's correct context, it will be seen to be quite a bit less shocking. When critics claim that "Mormon's teach Jesus and Satan are brothers", they are simply wrong. We do not teach that. It may be technically accurate to say, but it is not what I would call a part of our doctrine. You will never see an LDS sunday school lesson manual, or seminary lesson book with the slightest whisper of that idea. So it's simply not fair to claim that we teach it. The first time I ever heard the idea was when I was out tracting as a missionary and a non-member told me that I believed it. I answered, "Really? I've never even heard that before."
It's a bit like claiming George Washington and Adolf Hitler are brothers. It may be a true statement in the largest sense of the word 'brother' but it doesn't exactly throw any great light on their relationship. Members of the same human family yes, but beyond that, it's a stretch to refer to them as brothers.
According to LDS doctrine we all lived as spirits before we came into this life here on earth. In the pre-earth life, when we all lived with God in heaven, Satan or Lucifer, as he was then called (Isaiah 14:12) evidently held some high station. He was as Isaiah referred to him, a "son of the morning". I assume this may mean he was one of the early sons of God (whereas Jesus was the firstborn son of God). Lucifer however, rebelled against God and fought against God's plan, and was cast out of heaven.
We believe that God is our Eternal Father. He is the father, spiritually, of the whole human family. He created our spirits, and we are all his children. It is one of the first insights we gain from the Lord's prayer, when Jesus prayed "Our Father who art in heaven..."(Matt 6:9). Prophets have always taught this truth:
Forasmuch then as we are the offspring of God, we ought not to think that the Godhead is like unto gold, or silver, or stone, graven by art and man's device.
Jesus, in our theology is God's firstborn son, spiritually, and He was the only begotten in the flesh (that is, physically) as the scriptures teach.
For God so loved the world that He gave his only begotten Son that whosoever believeth in him should not perish, but have everlasting life.
He was the only begotten Son in mortality, but we all are the "offspring of God" spiritually.
Being made so much better than the angels, as he hath by inheritance obtained a more excellent name than they.
For unto which of the angels said he at any time, Thou art my Son, this day have I begotten thee? And again, I will be to him a Father, and he shall be to me a Son?
And again, when he bringeth in the firstbegotten into the world, he saith, And let all the angels of God worship him.
In other words, Jesus is the only one to whom God is both father spiritually and physically. Even the angels of God, who are his spiritual children, are not on the same level of exaltation as the Lord Jesus Christ. He alone was selected, as the firstborn spirit son and because of his perfection, to be the physical mortal Son of the Eternal Father.
He was chosen to be the literal offspring of God on the earth, the Savior and Redeemer of mankind. He was a God from the beginning, though until he completed his sacred mission of atonement he was not fully exalted and glorified as the Father is. He always was a member of the "Godhead" (Acts 17:29) comprised of the Father, the Son and Holy Ghost. It was through Jesus Christ that God created the earth (Hebrews 1:2). We believe that Jesus is the same being referred to as God and Jehovah in the Old Testament.
Lucifer, while once a spirit son of God and in some position of trust, was cast out of heaven, and was forever denied the opportunity to come to earth to obtain a physical body. He has never been physically born, and so Christ - and all of us - are a distinct level above Lucifer in reality. We are not on the same plane of glory, we kept our "first estate" while Satan and a third of the hosts of heaven did not.
And there was war in heaven: Michael and his angels fought against the dragon; and the dragon fought, and his angels,
And prevailed not; neither was their place found any more in heaven.
And the great dragon was cast out, that old serpent, called the Devil, and Satan, which deceiveth the whole world: he was cast out into the earth, and his angels were cast out with him.
How art thou fallen from heaven, O Lucifer, son of the morning! how art thou cut down to the ground, which did weaken the nations!
For thou hast said in thine heart, I will ascend into heaven, I will exalt my throne above the stars of God: I will sit also upon the mount of the congregation, in the sides of the north:
I will ascend above the heights of the clouds; I will be like the Most High.
Yet thou shalt be brought down to the sides of the pit.
They that see thee shall narrowly look upon thee, and consider thee, saying, Is this the man that made the earth to tremble, that did shake kingdoms...?
Through additional scriptures revealed to the Prophet Joseph Smith, we have even greater light and knowledge about what occurred in the beginning:
And I, the Lord God, spake unto Moses, saying: That Satan... is the same which was from the beginning, and he came to me, saying -- Behold, here am I, send me, I will be thy son, and I will redeem all mankind, that one soul shall not be lost, and surely I will do it; wherefore give me thine honor.
But, behold, my Beloved Son, which was my Beloved and Chosen from the beginning, said unto me -- Father, thy will be done, and the glory be thine forever.
Wherefore because that Satan rebelled against me, and sought to destroy the agency of man, which I the Lord God had given him, and also, that I should give unto him mine own Power; by the power of mine Only Begotten, I caused that he should be cast down;
And he became Satan, yea, even the devil, the father of all lies, to decieve and blind men, and to lead them captive at his will, even as many as would not hearken unto my voice.
Pearl of Great Price, Moses 4:1-4
Satan we are told here, desired the glory of God for himself, and presented an alternate plan. In his plan, no one would have freedom to make choices (agency), we would all be forced to keep his commandments and we would all return to heaven. In return for this, Satan desired God's throne, and his power. Unfortunately, his arguments were pursuasive, and he drew away a full 1/3 of the spirit children of God to follow him. This was the great war in heaven, and Satan and his followers (his angels), were cast out of heaven.
In summary, my answer to the original question is... no, Latter Day Saints don't believe that Jesus and Satan are brothers -- we believe they were brothers; they and all of God's children spiritually begotten by Him in the pre-earth life were brothers and sisters. However, because of Satan's rebellion from God and his being cast off for eternity, they can no longer be called brothers. Satan lost the grand inheritance he might have had, he sold it for pride and glory. He is no longer what he once was. He is no longer God's spirit son, but an enemy to God and his Only Begotten forever.
We all began as spiritual children of God; but there are children of God, and then there are children of God. The phrase has levels of meaning. Recorded in the 12th chapter of Matthew, the savior teaches us who his siblings are. In answer to an everyday request that he attend to his mother and brethren who "stand without, desiring to speak with thee." Jesus took the opportunity to teach this eternal truth:
But he answered and said unto him that told him, Who is my mother? and who are my brethren?
And he stretched forth his hand toward his disciples, and said, Behold my mother and my brethren!
For whoseover shall do the will of my Father which is in heaven, the same is my brother, and sister, and mother.
There are many other scriptures that echo this idea. Romans 8:16, 2 Cor. 6:18, 1 John 3:2, and numerous verses from the Book of Mormon (Ether 3:14, Moroni 7:19, Mosiah 5:7 for example) all speak about conversion and becoming truesons and daughters of God and Christ -- in the most complete sense. I finish with this one:
He came unto his own, and his own received him not.
But as many as received him, to them gave he power to become the sons of God, even to them that believe on his name.
For more information on this topic, see alsohttp://fairmormon.org/Jesus_Christ/Brother_of_Satan