I remember when I first got baptised and went the next day to talk to a local minister, that I knew well, about this. His response was to bring out this book with things against the church. He talked of Brigham Young and other people and events I knew nothing about.
At the time all I could think was that the Holy Ghost had told me to join the church and the Spirit was in it, and so it must be right, and there must be answers to these accusations. So I thought a rundown on who's who could be of value to some. If I've forgotten anyone significant feel free to comment and include them.
Before beginning I should define what "Mormonism" actually is. It refers to the truth: Those things Jesus Christ came and taught, and the Holy Ghost and Heavenly Father teach.
I could get technical and say that the highest position in the truth are eternal laws. But we'll keep it simple.
The highest position in the truth is that of the Father God. He is referred to as "Heavenly Father."
Secondly comes Jesus Christ, who is also a God; and referred to generally in Scripture as "Lord."
Thirdly comes The Holy Ghost. He isn't a God in the same sense, but reveals the Father God and the Lord Jesus. He teaches us and helps us create a relationship with Heavenly Father and Jesus Christ.
Next in position is us. All people are equal children of Heavenly Father and Heavenly Mother.
In addition to this we have the church that Jesus Christ set up for us, his brothers and sisters. The Church of Jesus Christ of Latter-Day Saints is part of the truth; as it sets out to teach those things that Heavenly Father and Jesus Christ would have for the majority of members to learn. Within this he has created a structured organisation. Meaning that there are offices within it for different functions.
The First Presidency is often composed of 3 men. Firstly we have what is termed "the President of the Church." Another term we use for him is "The Prophet." This term is used as he is the only one who is to receive revelation that is to be placed before the membership, sustained as revelation and placed within the Current Scripture. All members should be prophets for themselves etc. But this is a special prophet for the church as a whole.
His main role is as president, making operational decisions and encouragement to members.
The president has counselors who are also apostles (though not acting within the Council of Twelve Apostles - which comes next). They assist him in his role so that they can stand in his place where required (though they can't receive new doctrine for the church for sustaining).
The presidents of the church that we have had so far are (in order) _
Joseph Smith, Brigham Young, John Taylor, Wilford Woodruff, Lorenzo Snow, Joseph F. Smith, Heber J. Grant, George Albert Smith, David O. Mc Kay, Joseph Fielding Smith, Harold B. Lee, Spencer W. Kimball, Ezra Taft Benson, Howard W. Hunter, Gordon B. Hinckley and Thomas S. Monson.
Next comes the Council of 12 Apostles. There are always 12 in this council. They get assigned areas of responsibility within the church. Upon the death of "The Prophet" the next prophet becomes the apostle that has served as an apostle the longest period of time.
Additionally to this we have several other groups that also make up what is termed "General Authorities" (often referred to as GA's). Prophets and apostles also are GA's. These other groups have areas of responsibility within the church structure.
Arguably the "Standard Works" (Bible, Book of Mormon, Doctrine and Covenants, and Pearl of Great Price) hold the greatest authority within the church. I'm not saying "arguably" because it should be in dispute, but some are too lazy to read, ponder and pray. They lean on another of their mere mortal brothers to bring them to eternal life. Something Brigham Young said they should not be doing.
Church members generally place large authority to any comments made by Joseph Smith. Other "prophets" carry the next greatest weight. Particularly the "Current Prophet" gets special attention when comments of prophets are in disagreement.
This gets to a problem in quoting GAs (even past "Prophets"). If it doesn't suit it will be said that the "Current Prophet" hasn't said that.
"Prophets" will sometimes disagree with each other as not every word they say or acts they do are by revelation. We see examples of this in Scripture. For example Moses wouldn't circumcise his children as God commanded. His wife had to do it (Ex 4:25-26).
Some non-members feel that by demonstrating differences of opinion of past GAs that this somehow disproves "Mormonism." It must be remembered that the church isn't "Mormonism." "Mormonism" (the truth) is God's religion. It is based on eternal laws. The church is not a group of people who already have all the truth. The church is theoretically a group of people endeavouring to discover that truth more fully through revelation.