The Church of Jesus Christ of Latter-Day Saints does not practice polygamy. One man is allowed to have one wife, and vice-versa. This comes as a commandment from God. Often people confuse us with another church that broke off of our church many years ago, who also call themselves "Mormons." They practice polygamy, and have recently been in the spotlight on the news because of it. Unfortunately, I was serving my mission at the time, and I had so many people tell me that I practice polygamy. (it's funny, even after I would tell them we don't, they keep insisting, as if they know more about the church I belong to than I do! That was always humorous) Their confusion of course comes from the news coverage, with the media calling that group "Mormons." So naturally everyone thinks we do.
The answer, again, is no, we do not practice polygamy. It is true that in the past our church did practice plural marriages, by the command of God. There were times in the past as well during which God approved of one man having more than one wife, such as during the time of Abraham, (see Genesis 16:1-3) Jacob, (see Genesis 29:23-29) Gideon, (see Judges 8:30) and Elkanah (see 1 Samuel 1:1-2) to name a few.
It is interesting to note that God has told us that the commandment is one man shall have one wife. (see Jacob 2:27) He makes an exception to this commandment, however, in certain cases. (see Jacob 2:30)
The best source, of course, comes from one of our Modern Day Prophets, President Gordon B. Hinckley. He stated the following in the November 1998 Ensign:
"I wish to state categorically that this Church has nothing whatever to do with those practicing polygamy. They are not members of this Church. Most of them have never been members. They are in violation of the civil law. They know they are in violation of the law. They are subject to its penalties. The Church, of course, has no jurisdiction whatever in this matter.
If any of our members are found to be practicing plural marriage, they are excommunicated, the most serious penalty the Church can impose. Not only are those so involved in direct violation of the civil law, they are in violation of the law of this Church. An article of our faith is binding upon us. It states, “We believe in being subject to kings, presidents, rulers, and magistrates, in obeying, honoring, and sustaining the law” (A of F 1:12). One cannot obey the law and disobey the law at the same time.
There is no such thing as a “Mormon Fundamentalist.” It is a contradiction to use the two words together.
More than a century ago God clearly revealed unto His prophet Wilford Woodruff that the practice of plural marriage should be discontinued, which means that it is now against the law of God. Even in countries where civil or religious law allows polygamy, the Church teaches that marriage must be monogamous and does not accept into its membership those practicing plural marriage."
That is a Prophet of God telling us the will of God in these days. So there you have a complete answer. Do Mormons practice polygamy? No, we do not.