Since the assassination of Joseph Smith, Jr. on June 27, 1844 there has been a clear and established transfer of this prophetic calling to the next man, called of God, to serve in the capacity of prophet, seer and revelator.
Gordon B. Hinckley died on January 27, 2008 and the First Presidency was dissolved and the counselors returned to their positions in the Quorum of the Twelve Apostles, which now total fourteen. Until the new prophet is chosen, the Church will be run by the Apostles.
It has already been announced by the Church that the new prophet will not be chosen until the conclusion of President Hinckley’s funeral in the next few days.
When that time comes, the Apostles will gather together and through great prayer and inspiration will unanimously choose the next prophet, seer, revelator and president of the Church of Jesus Christ of Latter-day Saints.
From the Church’s website we learn:
The appointment of a new president of The Church of Jesus Christ of Latter-day Saints happens in an orderly way that — remarkably in today’s world — avoids any trace of internal lobbying for position or rank. Viewed by members as a divinely revealed process, it is devoid of electioneering whether behind the scenes or in public.
Moreover, it is not only the structure of Church organization that governs this process. There is also a deeply ingrained tradition in the Church that personal aspiration for leadership at any level is inappropriate. Instead, the emphasis is on personal worthiness and a humble willingness to serve when invited. (Succession in the Presidency of The Church of Jesus Christ of Latter-day Saints – www.lds.org Newsroom)
Traditionally it has been the president of the Quorum of the Twelve Apostles, who is Thomas S. Monson, called as the next prophet. Whomever that prophetic mantle falls on next, will then choose his counselors, generally from among the Quorum of Twelve Apostles. The new president of the Quorum of the Twelve Apostles, along with the rest of the quorum, will then lay their hands upon the head of the new prophet, seer and revelator and through a special prayer known as a blessing, set him apart as the new Prophet and President of the Church of Jesus Christ of Latter-day Saints.
Since Brigham Young was called to follow Joseph Smith, Jr. as the prophet of the Church, it has always been conducted in a mannerly fashion. Gordon B. Hinckley was the 15th prophet of the Church called in these modern times.