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?