Mormons largely saw the Civil War as punishment being meted out to the United States for their treatment of their faith in the years leading up to it. By the time of the Civil War began in 1861, the majority of Mormons had traveled west and settled in Utah, which at that time was US territory and called Utah Territory. This group, often referred to as the LDS Church or the Brighamite Church, as it was led by Brigham Young, had a largely negative view of the United States government. Just four years prior to the beginning of the Civil War, the Mormons were involved in the Utah War. Much can be said about that conflict, but essentially it was a conflict between the appointed administrators and judges of the Utah territory who were non-Mormon, and the populace, which was overwhelmingly Mormon and preferred to recognize their religious leaders as having authority over almost everything. As these administrators reported back to Washington and then some left their posts for fear of their safety, President James Buchannon sent troops to Utah, fearing rebellion. The Mormons in turn prepared for the troops, which they had learned were coming but whose intentions they did not know. They prepared for evacuation (and actually did leave Salt Lake City before troops arrived) and made plans to burn their homes, crops, and property so the US troops would have no resources to stay.
The result of the Utah War was that Buchannon replaced Brigham Young as governor and left a garrison of US troops to enforce civil authority in Utah, which was immensely resented by the Mormons and described as an occupation.
In that context, and with the prior grievances of Mormons stemming from conflicts in Missouri and Illinois, the Civil War was seen as an affliction upon all of the United States, and one that was not within the Mormon's ability to control or influence. As a result, the Mormons largely did not participate in the war. Their only real involvement was in raising a couple groups of Calvary to maintain trade/mail routes safe from Indian raids when the US troops left as a result of the need for all troops to engage in the Civil War.
The best estimate of Mormon participants in the Civil War indicate that only a few hundred Mormons participated, mostly for the North, as the majority of Mormons had originally been from the Northeast of the United States. Mormon men were not encouraged to fight in the war.
Most leaders who expressed opinions on the matter indicated that the war was God's punishment to the United States for its treatment of the church, both in Missouri, from which it had been expelled, and then in connection with the Utah War. One leader of the church, Orson Hyde, during the Civil War, declared that "Joseph Smith once said, on the stand in Nauvoo, Illinois, that if the Government of the United States did not redress the wrongs of the Mormon people inflicted upon them in the State of Missouri, the whole nation should be distracted by mobs from one end to the other; and that they should have mobs to the full and to their ‘hearts content.’" Another leader, Wilford Woodruff, speaking in 1862 about the Civil War declared, "It is a hard dealing of the Almighty and we cannot help it.”
Brigham Young also declared it to be a scourge on the United States for its treatment of the church and its members. “God has come out of his hiding place and has commenced to vex the nation that has rejected us, and he will vex it with a sore vexation. It will not be patched up—it never can come together again—but it will be sifted with a sieve of vanity, and in a short time it will be like water spilled on the ground.”

During the War, the church and its militia, the Nauvoo Legion largely stayed out of the war. In 1862, during the Civil War, the United States passed the Morrill Anti-Bigamy Act in response to the polygamy practiced by Mormons at that time, as well as the massive ownership of property by the Mormon church. The law banned polygamy in US territory and also prohibited non-profit ownership of land in US territory over $50,000. According to the book Zion in the Courts by Edwin Brown and Richard Mangrum (2001), Abraham Lincoln chose not to enforce the law in exchange for Utah's non-participation in the Civil War. "Abraham Lincoln reportedly compared the LDS Church to a log he had encountered as a farmer that was 'too hard to split, too wet to burn and too heavy to move, so we plow around it. That's what I intend to do with the Mormons. You go back and tell Brigham Young that if he will let me alone, I will let him alone.'"
So the Mormon church did not pick sides, and rather stayed out of the conflict and rather saw it as a conflict between those that had afflicted them. For more reading, see the book above and also “We Know No North, No South, No East, No West”: Mormon Interpretations of the Civil War, 1861–1865 by Richard E. Bennett.