Polygamy is the practice of having more than one wife and is an older doctrine within the Mormon religion. It involved taking more than one wife, an accusation pointed at both Joseph Smith and Brigham Young, the earliest of Mormon prophets. Although the practice of polygamy before the Millenial Kingdom (1000 year reign) is not an active doctrine in most Mormon circles today, it set Mormons apart from the rest of the United States during the 19th century. There are accounts indicating that Joseph Smith himself had 27 wives2 though he publicly denied this many times (cf. the official Mormon family geneology web site). Because of his denials, many of the splinter groups headquartered in the eastern and southern states do not believe Joseph Smith ever taught the practice. If they do acknowledge polygamy as a teaching of Joseph Smith, they claim that it was a sin on Smith's part and was not a true principle given from God. These groups do not believe many of the doctrines taught by Smith in the last few years of his life, claiming that he had fallen away from the truth.
It was Brigham Young (with 56 wives)3 who, in 1852, first gave polygamy the push to become a semi-regular practice within the LDS Church in Utah. After a series of legal decisions made by the United States government toward the late 1800's, the church issued an official declaration  suggesting that members refrain from any marriage that is unlawful in the land. The practice continued for several years in Canada and Mexico, but gradually came to an end. Today, members of the LDS Church are excommunicated if they are found to be practicing polygamy. There are several splinter groups in Utah who encourage the practice of polygamy while sharing many beliefs and doctrines with the LDS Church.
The doctrine continues today in subtle form. If a wife dies a husband may be sealed in the temple with another wife without annulling the previous sealing. Thus, the man is expected to spend eternity in union with at least those two wives. Some mainstream Mormons also hold that the practice will continue in the Millenial Kingdom. Some, but few, still assert that plural marriage is necessary for Celestial exaltation (as attested to by Brigham Young). Given that there are more women than men required in this doctrinal framework, some hold that there will be more females in the Celestial kingdom, and that females are somehow more spiritually inclined than men, thus proving themselves worthy of their entrance into the kingdom.
It is also of note that D&C 132 is still a part of the Mormon canon.
 Polygamy is condemned in the Book of Mormon
In the Book of Mormon's book of Jacob, chapter 2, verses 23-29 and 33-35, the practice of taking multiple wives is described as "whoredoms" and "abominable" before God, and the lands that practice such things are cursed by God.
 Necessary for exaltation
We know that marriage is required for exaltation:
- "From [Joseph Smith] I learned that the doctrine of plural and celestial marriage is the most holy and important doctrine ever revealed to man on the earth, and that without obedience to that principle, no man can ever attain to the fulness of exaltation in celestial glory." - William Clayton, Historical Record, Vol. 6, pp. 225-227
"Our exaltation depends on marriage.... Our Heavenly Father has given us the law of eternal marriage so that we can become Like him. We must live the law of eternal marriage to become as he is -- able to have spirit children." (Gospel Principles, p. 231.)
How polygamy factors in to exaltation is commonly misunderstood, and it is in fact actively obfuscated by modern LDS leaders, teachers, and apologists (commonly discounted with a reference to Official Declaration 1).
LDS Church teachers in time since D&C 132, have demanded that polygamy be followed, and reinforce the damnation of not entering into polygamy:
". . . he that abideth not this law (polygamy) can in no wise enter into my glory, but shall be damned, saith the Lord." (D & C, 132:27)
"Go ye, therefore, and do the works of Abraham; enter ye into my law [polygamy] and ye shall be saved." (D & C, 132:32)
'"The only men who become Gods . . . are those who enter into polygamy." (Brigham Young, Journal of Discourses, vol. 11, p. 269)
"I bear my solemn testimony that plural marriage is as true as any principle that has been revealed from the heavens. I bear my testimony that it is a necessity, and that the Church of Christ in its fullness never existed without it. Where you have the eternity of marriage you are bound to have plural marriage; bound to and it is one of the marks of the Church of Jesus Christ in its sealing ordinances." (George Teasdale, Journal of Discourses, vol. 25, p. 21)
'"The Lord has said, that those who reject this principle [polygamy] reject their salvation, they shall be damned, saith the Lord . . . they will finally go down to hell and be damned if they do not repent." (0rson Pratt, Journal of Discourses, vol. 17, pp. 224-225)
 Faith-promoting rumors of early Mormon demographics
Although many Mormons defend "the practice of polygamy in the early days of the Church by pointing to a surplus of women in Utah, census reports for the time show roughly equal numbers of men and women." 
 D&C 101 (since removed)
This was removed in 1876 and replaced with section 132.
"Inasmuch as this Church of Christ has been reproached with the crime of fornication and polygamy, we declare that we believe that one man should have one wife, and one woman but one husband, except in the case of death, when either is at liberty to marry again." -D&C 101:4, History of the Church, vol. 2, pg. 247
 After the 1890 Manifesto
"After the 1890 Manifesto officially ended plural marriage, thousands of Mormons in good standing, including a number of apostles, continued to marry multiple wives in secret. When Congress demanded that furtive polygamists be rooted out, Joseph F. Smith, who was a nephew of the founder and became president of the Church, issued a Second Manifesto, in 1906, in which he declared that anyone who participated in the practice would be excommunicated. Nonetheless, Smith himself continued to perform secret plural marriages. In 1933, the Church's president, Heber J. Grant, began a determined policy to eradicate polygamy altogether." 
Richard S. Van Wagoner, Sidney Rigdon: Portrait of Religious Excess, p. 293:
A multitude of Mormon records provides irrefutable evidence for Smith's prerogative with an array of women, many of them just a few years older than his own children. And while the prophet now stands astride the Mormon world like a colossus, in Nauvoo he maneuvered within the charisma of his own mystique to defy both church, Nauvoo City, and Illinois marriage laws, as well as to conceal his behavior from his wife Emma. This equivocal deportment, secreted by a deferential and circumspect group of men and women, created two cultures in Nauvoo—one where monogamy and fidelity prevailed—the other where eros and duplicity seemed to subvert the highest moral values, and where exonerating the "Lord's Anointed" became more important than telling the truth.
This dichotomy left Joseph's and Emma's marriage hanging by a thread. Emma spent the last three years of her husband's life jealously battling his errant yearnings, more than once threatening to return to her family in New York. On one occasion, according to Smith's private secretary, she threatened that if he continued to "indulge himself she would too." Although Emma apparently countenanced two of her husband's 1843 sealings—to Emily and Eliza Partridge—she recanted within a day and demanded that Joseph give them up or "blood should flow." Her change of heart came after she found Joseph and Eliza Partridge secluded in an upstairs bedroom at the Smith home. The realization that the sealing represented more than a "spiritual marriage" or "adoptive ordinance" devastated her.
Smith used this ruse that same month, May 1843, to convince another young woman, Helen Mar Kimball, that her sealing to him would be of a "spiritual order and not a temporal one." Helen, fifteen-year-old daughter of Apostle Heber C. Kimball, reported that the prophet admonished her: "If you will take this step, it will insure your eternal salvation & exaltation and that of your father's household & all of your kindred." "This promise was so great," Helen later remembered, "that I willingly gave myself to purchase so glorious a reward." Lamenting her [p.294] decision, Helen confided to a close Nauvoo friend: "I would never have been sealed to Joseph had I known it was anything more than ceremony. I was young, and they deceived me, by saying the salvation of our whole family depended on it."
- "The doctrine of polygamy with the Mormons is not one of that kind that in the religious world is classed with 'non essentials.' It is not an item of doctrine that can be yielded, and faith in the system remain.... Mormonism is that kind of religion the entire divinity of which is invalidated, and its truth utterly rejected, the moment any one of its leading principles is acknowledged to be false..... Polygamy was revealed by God, or the entire fabric of their faith is false. To ask them to give up such an item of belief, is to ask them to relinquish the whole, to acknowledge their Priesthood a lie, their ordinances a deception, and all that they have toiled for, lived for, bled for, prayed for, or hoped for, a miserable failure and a waste of life." Millennial Star - Vol 5:27, October 1865
- "God will not change his law of celestial marriage. But the man, the people, the nation, that oppose and fight against this doctrine and the Church of God will be overthrown." Lorenzo Snow (1886, from jail) - History of Utah, Whitney, 3:471
- "Some people have supposed that the doctrine of plural marriage was a sort of superfluity, or non-essential to the salvation of exaltation of mankind. In other words, some of the Saints have said, and believe, that a man with one wife, sealed to him by the authority of the Priesthood for time and eternity, will receive an exaltation as great and glorious, if he is faithful, as he possibly could with more than one. I want here to enter my solemn protest against this idea, for I know it is false."
Joseph F. Smith (Prophet, Seer and Revelator) JD 20:28
- "Those who made the attack perhaps hope to drive the people of God to renounce the doctrine and promise not to obey the revelation. Unless the Saints apostatize, such an action on their part is impossible. By doing so, they would deliberately shut the door of the celestial glory in their own faces.... To comply with the request of our enemies would be to give up all hope of ever entering into the glory of God, the Father, and Jesus Christ, the Son. ... So intimately interwoven is this precious doctrine with the exaltation of men and women in the great hereafter that it cannot be given up without giving up at the same time all hope of immortal glory." George Q. Cannon - Jun. Instructor, May 1, 1885, Editorial
- "The implied assumption in this theory, that there have been more female than male members in the Church, is not supported by existing evidence. On the contrary, there seems always to have been more males than females in the Church... The United States census records from 1850 to 1940, and all available Church records, uniformly show a preponderance of males in Utah, and in the Church. Indeed, the excess in Utah has usually been larger than for the whole United States... there was no surplus of women." - Apostle John A. Widstoe, Evidences and Reconciliations, pp. 390-392
- "... in the Utah period the number who lived in plural households was considerably larger than previously believed. During the 1880s, Mormon representatives in testimony before Congress stated that no more than 1 or 2 percent of the church’s membership was polygamous. Church authorities in their sermons, missionaries abroad, and guides on Temple Square almost to the present time have repeated these figures. We now know, owing to work by [Larry] Logue, [Ben] Bennion and others, that the actual number, depending on the years and location, likely averaged between 15 and 30 percent."
- "The cases that are to be handled by the Church include but are not limited to fornication, adultery, homosexual acts, abortion, or other infractions of the moral code; intemperance; criminal acts involving moral turpitude, such as burglary, dishonesty, theft, or murder; apostasy; open opposition to and deliberate disobedience of the rules and regulations of the Church; cruelty to spouse or children; advocating or practicing so-called plural marriage; or any un-Christianlike conduct in violation of the law and order of the Church." - President N. Eldon Tanner, First Counselor in the First Presidency, "Priesthood Responsibilities", April 1973 Conference Report
- "REUTERS. Turning to polygamy. I realize the church banned the practice in 1890, but is it true that LDS temples across the world continue to conduct ceremonies where men are sealed to multiple wives who would only become their spouses in the afterlife? Does that happen? CHRISTOFFERSON: Yes, when someone's spouse has died for example and they remarry; they could be married in a temple for a second time, sealed as we say. How that sorts out in the afterlife we'll leave in the Savior's hand, but we permit people to marry again when there's been a death or a divorce."
- "Another theory is that Joseph married polyandrously when the marriage was unhappy. If this were true, it would have been easy for the woman to divorce her husband, then marry Smith. But none of these women did so; some of them stayed with their 'first husbands' until death. In the case of Zina Huntington Jacobs and Henry Jacobs---often used as an example of Smith Marrying a woman who's marriage was unhappy---the Mormon leader married her just seven months after she married Jacobs and then she stayed for years after Smith's death. Then the separation was forced when Brigham Young (who had married Zina polyandrously in the Nauvoo temple) sent Jacobs on a mission to England and began living with Zina himself." - Todd Compton, In Sacred Loneliness
- "Partly to maintain secrecy, Joseph could not have spent much time with [Louisa] Beaman or any of the women he married. He never gathered his wives into a household--as his Utah followers later did--or accompanied them to public events. Close relationships were further curtailed by business. Joseph had to look after Emma and the children, manage the Church, govern the city, and evade the extradition officers from Missouri. As the marriages increased, there were fewer and fewer opportunities for seeing each wife. Even so, nothing indicates that sexual relations were left out of plural marriages" (Richard Lyman Bushman, Joseph Smith: Rough Stone Rolling [New York: Knopf, 2005], 438-39).
- "Meanwhile, the Prophet, with Louisa Beeman and my sister Delcena, had it agreeable arranged with Sister Almera, and after a little instruction she stood by the Prophet's side and was sealed to him as a wife, by Brother Clayton; after which the Prophet asked me to take my sister to occupy number "10" in his Mansion home during her stay in the city. But as I could not long be absent from my home and business, we soon returned to Ramus, where on the 15th of May, some three weeks later, the Prophet again came and at my house occupied the same room and bed with my sister, that the month previous he had occupied with the daughter of the late Bishop Partridge, as his wife." - Benjamin F. Johnson, Letter to George S. Gibbs, 1903, cited in E. Dale LeBaron, "Benjamin Franklin Johnson: Colonizer, Public Servant, and Church Leader" (M.A. thesis, Brigham Young University, 1967)