Short answers, pulled from the story.
A sacrament is a Christian rite recognised as particularly important and significant. Many denominations follow the definition formulated by Augustine of Hippo: an outward sign of an inward grace, instituted by Jesus Christ.
The Catholic Church recognises seven sacraments: Baptism, Confirmation, the Eucharist, Penance, Anointing of the Sick, Holy Orders, and Matrimony. The Council of Trent (1545-1563) reaffirmed this list and declared that holding any other number was grounds for anathema.
Most Protestant denominations, including Reformed, Anglican, and Methodist churches, recognise two sacraments ordained by Christ: Baptism and the Lord's Supper. Lutherans often add Confession and Absolution as a third sacrament.
The English word derives from the Ecclesiastical Latin sacramentum, which in Ancient Rome meant a soldier's oath of allegiance. The third-century writer Tertullian drew a parallel between that oath and Christian initiation through baptism and the Eucharist.
Ex opere operato means the sacrament is effective by the very fact of being performed, regardless of the personal holiness of the minister administering it. The Catholic Church holds this position, and a similar view appears in Article XXVI of the Anglican Thirty-Nine Articles.
Yes. The Salvation Army does not practice formal sacraments, preferring to focus on the reality behind the symbols. The Quakers (Religious Society of Friends) believe all of life can be a means of God's grace and perform no outward sacramental rites. The sixteenth-century Anabaptist group the Clancularii also rejected outward sacraments entirely.