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A district is an organizational unit of the church comprised of several small congregations or branches. Districts are presided over by a District President under the direction of a Mission President. This is not to be confused with a missionary district.
Prior to the 1920s districts were know as Conferences. In his Encyclopedic History of the Church of Jesus Christ of Latter-day Saints, Andrew Jenson uses the term "conference" but admits that by then the term district was becoming the preferred form.
Even though districts are a state of church organization indicating the church is not as fully organized as where there are stakes, districts still indicate the church has progressed to some maturity in the area.
This is especially so currently, since the general policy is that districts should be lead by local brethren. If it is not felt that any local brethren have the spiritual maturity to run a district, the area is generally administered directly by the mission.