How is the leadership of the Church organized?

       The Church of Jesus Christ of Latter-day Saints is governed by two main bodies with equal power: the First Presidency and the Quorum of the Twelve Apostles.
       The First Presidency is comprised of the President of the Church, who is also the Prophet, and his two counselors. He is the only man on earth who holds all of the keys ("keys" are how we describe the permission and authority from God) for administering the Church. The fact that there are three members of the First Presidency is symbolic of the Godhead (Heavenly Father, Jesus Christ, and the Holy Ghost).
       The Quorum of the Twelve Apostles (sometimes unofficially referred to as simply "the Brethren"), is composed of twelve men whose primary mission it is to be special witnesses of Jesus Christ and to oversee the different departments and programs of the Church. They also do a lot of other stuff, as do the members of the First Presidency. Running a world-wide church is a lot of work!
       The members of the First Presidency and the Quorum of the Twelve Apostles are not elected, but are chosen by the First Presidency and/or the Quorum of the Twelve Apostles when one of its members dies. They pray about it and God reveals to them who the new guy should be. Also, the Q of the 12 operates by consensus. They don't make decisions unless they come to a consensus. That's why it's so important for all of the members to be righteous, spiritually in-tune people. They must be humble and always ask God what He wants done for His church. If they don't repent of their sins, they can't receive revelation.
      Most importantly, the First Presidency and the Quorum of the Twelve are just a bunch of men through whom God acts. We don't worship them or anything like them, but we do respect them a lot. They are God's servants, and they do for the Church only the things that God wants them to do.