Companion(s): Every missionary is assigned a fellow missionary to be with at all times. These missionaries are called companions. They do everything together for the time they are companions. See "Transfers."
Elder: What male missionaries are called. Also, apostles are referred to as Elders.
Mission: a "trip," usually for 18-24 months, where the Missionary dedicates 100% of their time to serving the Lord through service, teaching, preaching, etc.
Mission Leadership Positions: The Mission President is in charge of the mission. He and his wife are often viewed as parental figures for the missionaries. The rest of the leadership positions are filled by regular missionaries within the mission, and the missionaries who fill them change periodically. Assistants to the President (APs) do just that: they assist the president in any and every way he asks them to. Zone Leaders (ZLs) are under the APs, and supervise a number of companionships. There are a couple of Zone Leaders under each AP. District Leaders (DLs) are under Zone Leaders and usually supervise 2-4 companionships. There are a few District Leaders under each Zone Leader.
Transfers: every 6-8 weeks, when (under the direction of the Mission President) many missionaries change companions and/or switch areas within their mission boundaries (ie moving to different "suburbs" in the western Santiago metropolitan area).