Federal headship refers to the representation of a group united under a federation or covenant. For example, a country's president may be seen as the federal head of their nation, representing and speaking on its behalf before the rest of the world.
In Christianity
In Christianity, this concept has been used to explain the concepts of the covenants found in the Bible. In particular, it has been applied to passages such as Romans 5:12-21, explaining the relation of all humanity with Adam, as well as the relation of redeemed humanity with Jesus Christ, who is called the last Adam. According to this understanding, as humanity's federal head Adam brought the entire human race into sin, misery, and death due to his disobedience. Christ, in his perfect obedience to God the Father, earned eternal life and blessedness for all his people.
These concepts can be found in the writings of the Church Fathers, including Irenaeus' Against Heresies and Augustine's City of God. The full theological articulation came in the time of the Protestant Reformation, and this doctrine is held by many Protestant churches, particularly in Presbyterian and Reformed churches, but not all.