In the Latin Church, it is ordinarily understood that the spouses, as ministers of Christ's grace, mutually confer upon each other the sacrament of Matrimony by expressing their consent before the Church. In the Eastern liturgies the minister of this sacrament (which is called "Crowning") is the priest or bishop who, after receiving the mutual consent of the spouses, successively crowns the bridegroom and the bride as a sign of the marriage covenant.
Source: Catechism of the Catholic Church (Vatican.va)