There is no other church on earth like The Church of Jesus Christ of Latter-day Saints. Its organization and growth are inspired. Much of the organization of the church is built upon the following principle: "No man taketh this honour unto himself, but he that is called of God, as was Aaron" (Heb. 5:4). The LDS Church is largely run by a lay ministry. People do not vie for positions; most do not seek office. When a call is received, service begins. The LDS church is organized in such a manner that all have the opportunity in even a small way to play a role in the church. The church is run and directed by and through the priesthood of God. In this essay, I'll provide an overview of how the LDS Church is organized.
I'll start from the "top" and move down. Actually this is starting at the bottom - at the foundation - and moving up, but we more commonly think of it as top-down. The Savior Jesus Christ is the head of the LDS Church. It is His church and is a restoration of the church He established in ancient days. The Savior chooses one man to act as His prophet - to preside over and to lead the church on earth. The current prophet is Thomas S. Monson, who is the most senior apostle and is ordained as a prophet, seer, and revelator. He is the only one authorized to speak definitively for the whole church and to establish church doctrine. He can delegate this authority in specific circumstances but generally does not. The prophet is the presiding priesthood holder on the earth. He holds all of the keys of the kingdom, meaning that he has the authority and responsibility to direct the church as inspired. The prophet has two counselors (although there could be more) who as a whole constitute the First Presidency of the LDS Church. The First Presidency together also hold all the keys of administration of the church. They constitute the highest governing body of the church and serve in a small way as a type for the Godhead - the Father, Son, and Holy Ghost (in other words, there are three people in the First Presidency in part to mirror the three personages in the Godhead).
The members of the First Presidency are all prophets, seers, and revelators in addition to being ordained as apostles. While they are currently called from the Quorum of the Twelve Apostles, their roles are separate from the Quorum of the Twelve while they are in the First Presidency. Upon the death of the prophet, the First Presidency is automatically dissolved and the counselors return to their place in the Quorum of the Twelve.
The Quorum of the Twelve consists of twelve men who are ordained as apostles of the Lord Jesus Christ and who are sustained as prophets, seers, and revelators. They are all special witnesses of the Lord Jesus with a special call to bear witness of Him unto all the world. They also serve as the main, under the direction of the First Presidency, administrative body of the church. One man is called as president of the Quorum (with another called as acting president should the president of the Twelve be called as a counselor in the First Presidency). The Twelve, along with the First Presidency, fulfill different assignments throughout the world, including dedicating new temples, meeting with church members, meeting with world, religious, and civic leaders, meeting with the media, and so forth. Each also serve on various committees pertaining to different aspects of church administrations, such as missionary work or education. The Twelve each individually hold all the keys of the priesthood but are not individually or collectively authorized to use those keys except as following the death of the prophet; in that instance, they can only use their keys collectively. The members of the Quorum of the Twelve are "ranked" according to seniority based on how long they've been a member of the quorum. The most senior apostle, following the death of the prophet, is shortly sustained as the new prophet and president of the church. There is no jostling for position or asking of questions about who will be called. The new prophet then calls two counselors. Any voids left in the Quorum of the Twelve are then subsequently filled with the callings and sustainings of new apostles.