Eternal Rest

(1) to hasten the progression of the souls of the faithful departed in Purgatory to their place in Heaven (in Catholicism)
(2) to rest in the love of God the souls of the faithful departed in Paradise until the resurrection of the dead and Last Judgement (in Catholicism, Lutheranism, Anglicanism and Methodism)

The prayer is cited from 2 Esdras (4 Esdras Vulgate):

Therefore, I say to you, O nations that hear and understand, “Wait for your shepherd; he will give you everlasting rest, because he who will come at the end of the age is close at hand. Be ready for the rewards of the kingdom, because perpetual light will shine on you forevermore. -2 Esdras 2:34-35 NRSV

Theology

This Catholic doctrine is found in the Catechism of the Catholic Church, paragraphs 1030-1032:

All who die in God's grace and friendship, but still imperfectly purified, are indeed assured of their eternal salvation; but after death they undergo purification, so as to achieve the holiness necessary to enter the joy of heaven. The Church gives the name Purgatory to this final purification of the elect, which is entirely different from the punishment of the damned...From the beginning the Church has honoured the memory of the dead and offered prayers in suffrage for them, above all the Eucharistic sacrifice, so that, thus purified, they may attain the beatific vision of God. The Church also commends almsgiving, indulgences, and works of penance undertaken on behalf of the dead.

The Lutheran cleric Richard Futrell wrote that "The historic practice within the Lutheran Church had prayers for the dead in their Prayer of the Church. For example, if we were to look at a typical Lutheran service during Luther’s lifetime, we would find in the Prayer of the Church not only intercessions, special prayers, and the Lord’s Prayer, which are still typical today in Lutheran worship, but also prayers for the dead." For those who have died, Martin Luther declared that 'I regard it as no sin to pray with free devotion in this or some similar fashion: Dear God, if this soul is in a condition accessible to mercy, be thou gracious to it. (Luther’s Works, Volume 37)

The United Methodist Church teaches the "truth of intercessory prayer for the dead" and that "prayer for the dead has been a widespread practice throughout Christian history [and] is a profound act of love addressed to a God of love".