In the LDS church what is the process of calling a new prophet? The Church of Jesus Christ of Latter Day Saints was originally founded by Joseph Smith believed to be a prophet of God. After his death he was followed by Brigham Young and after Brighams death it was followed by John Taylor. 

This pattern of a prophet dying and then a new one being called appears to be repeated again and again in the LDS church. 

What is the usual process that is followed for calling a new prophet?

How is the decision made and by whom?

Were there any prophets called by a different process or unusual circumstances?
The Church had been initially organized in 1830 with Joseph Smith as the first Elder and Oliver Cowdery as the second. In 1832, Joseph Smith organized a First Presidency and appointed two counselors. One of these was Sidney Rigdon, The other was excommunicated in 1833 and his replacement disappeared about 1837. Oliver Cowdery had been appointed Assistant President; when he was excommunicated in 1838, Joseph Smith's brother Hyrum Smith was appointed to the position. Other counselors had been appointed, but most of these had either died or left the church.

Following the death of Joseph Smith, it was not entirely clear who should lead  the church. One of the two principal claimants at a Church conference called within a few months was Sidney Rigdon, who made a claim based on his long service as counselor to and spokesman for Joseph Smith, although he had become somewhat disassociated from the main body of the Church. The other was Brigham Young, who claimed that the office of leadership should have gone to Hyrum Smith who had been killed along with Joseph, but that the next highest authority was the Twelve apostles, of which he was the senior and president. The conference accepted the leadership of the apostles, and rejected that of Sidney Rigdon.  Three years later, in 1847 after returning from the pioneering expedition to the Salt Lake Valley to Iowa where most of the members of the Church were in temporary quarters, Brigham Young proposed reorganization of the First Presidency, with himself as the President of the Church. 

Since that time, established practice and doctrine has it that the President of the Church is the supreme authority in the Church, below Christ himself. He is assisted by two (or occasionally more) counselors whom he selects. These form a First Presidency and are the highest governing body. 

The next highest body is the Quorum of the Twelve Apostles. These are sustained and accepted by members of the Church as "Prophets, Seers, and Revelators" when they are called and ordained as apostles.

When a President of the church dies, the First Presidency is dissolved and the counselors resume whatever position they had prior to their call to the First Presidency. They have almost always been apostles and members of the Quorum of the Twelve Apostles, but there have been a few exceptions. This quorum is then the interim highest governing body, until such time as they, after prayer and by revelation, decide to reorganize the First Presidency. The man chosen as President of the Church has always been the senior living apostle (as determined by continuous length of service). It would be a major departure from the established order for anyone else to be named.