The exhortation Orate Fratres (the original Latin), "Pray, brethren (brothers and sisters), that my sacrifice and yours be acceptable to God the Father almighty" is addressed by the celebrant to the people before the Prayer over the Offerings in the Roman Rite Mass, as well as in the Anglo-Catholic Mass. It is answered: "May the Lord accept the sacrifice at your hands for the praise and glory of his name, for our good and the good of all his holy Church."
In the Tridentine Mass, the celebrant then adds: "Amen".
The priest's Orate, fratres is merely a medieval amplification of the usual Oremus before any prayer. The Jacobite rite has an almost identical form before the Anaphora (Brightman, "Eastern Liturgies", Oxford, 1896, 83); the Nestorian celebrant says: "My brethren, pray for me" (ib., 274). Such invitations, often made by the deacon, are common in the Eastern rites. The Gallican rite had a similar one (Duchesne, "Christian Worship", London, 1904, 109). The Mozarabic invitation at this place is: "Help me brethren by your prayers and pray to God for me" (P.L. LXXXV, 537). The medieval derived rites had similar formulæ (e. g. "Missale Sarum", Burntisland, 1861-3, 596). Many of the old Roman Secrets (really Offertory prayers) contain the same ideas. Durandus knows the Orate Fratres in a slightly different form ("Rationale", IV, 32). A proof that it is not an integral part of the old Roman Mass is that it is always said, not sung, aloud (as also are the Tridentine Mass prayers at the foot of the altar, last Gospel etc.).
In the Tridentine Mass rubrics, after the "Suscipe Sancta Trinitas", the priest kisses the altar, turns to the people and says aloud: Orate fratres, extending and joining his hands. Turning back he finishes the sentence inaudibly. At high Mass the deacon or subdeacon, at low Mass the server, answers. The rubric of the Missal is: "The server or people around answer, if not the priest himself." In this last case he naturally changes the word tuis 'your' to meis'my'.
This article incorporates text from a publication now in the public domain: Herbermann, Charles, ed (1913). "Orate Fratres". Catholic Encyclopedia. Robert Appleton Company.