The first Quorum of the Twelve was called in February 1835 at Kirtland, Ohio. The three surviving witnesses to the Book of Mormon, Oliver Cowdery, David Whitmer and Martin Harris, chose the men, and the quorum's original charge was missionary rather than governing: it was a travelling council with authority in the world at large, while the organised stakes of the church were run by high councils. That distinction mattered at the time and disappeared entirely later.
Its formative undertaking was the mission to Britain of 1839 to 1841, during which the quorum, several of its members seriously ill on arrival, brought in several thousand converts and established the emigration route that reshaped the movement's demography for the rest of the century. It also established Brigham Young as the quorum's effective leader.
The 1830s were unstable in a way that the quorum's own composition records plainly: nine of the original twelve either left the movement or were excommunicated, several of them within five years. That figure is the best single measure of how uncertain the early church was.
Everything changed at Nauvoo in August 1844. With Joseph Smith dead and Sidney Rigdon claiming the guardianship of the church, the assembled Saints sustained the Quorum of the Twelve as the presiding authority. The quorum led the church through the exodus and governed alone until Brigham Young was ordained president in December 1847, and it has been the second body of the church ever since.