The rite by which the bread and wine are presented (offered) to God before they are consecrated, and the prayers and chant that accompany it.
The idea of this preparatory hallowing of the matter of the sacrifice by offering it to God is very old and forms an important element of every Christian liturgy. In the earliest period we have no evidence of anything but the bringing up of the bread and wine as they are wanted, before the Consecration prayer. Justin Martyr says: "Then bread and a cup of water and wine are brought to the president of the brethren" (I Apol., lxv, cf, lxvii). But soon the placing of the offering on the altar was accompanied by a prayer that God should accept these gifts, sanctify them, change them into the Body and Blood of his Son, and give us in return the grace of Communion. The Liturgy of "Apost. Const." VIII, says: "The deacons bring the gifts to the bishop at the altar . . . (xii, 3-4). This silent prayer is undoubtedly an Offertory prayer. But a later modification in the East brought about one of the characteristic differences between Eastern and Roman liturgies. All Eastern (and the old Gallican) rites prepare the gift before the Liturgy begins. This ceremony (proskomide) is especially elaborate in the Byzantine and its derived rites. It takes place on the credence table. The bread and wine are arranged, divided, incensed; and many prayers are said over them involving the idea of an offertory. The gifts are left there and are brought to the altar in solemn procession at the beginning of the Liturgy of the Faithful. This leaves no room for another offertory then. However, when they are placed on the altar prayers are said by the celebrant and a litany by the deacon which repeat the offertory idea. Rome alone has kept the older custom of one offertory and of preparing the gifts when they are wanted at the beginning of the Mass of the Faithful. Originally at this moment the people brought up bread and wine which were received by the deacons and placed by them on the altar. Traces of the custom remain at a papal Mass and at Milan. The office of the vecchioni in Milan cathedral, often quoted as an Ambrosian peculiarity, is really a Roman addition that spoils the order of the old Milanese rite. Originally the only Roman Offertory prayers were the secrets. The Gregorian Sacramentary contains only the rubric: "deinde offertorium, et dicitur oratio super oblata" (P.L. LXXVIII, 25). The Oratio super oblata is the Secret. All the old secrets express the offertory idea clearly. They were said silently by the celebrant (hence their name) and so are not introduced by Oremus. This corresponds to the oldest custom mentioned in the "Apost. Const."; its reason is that meanwhile the people sang a psalm (the Offertory chant). In the Middle Ages, as the public presentation of the gifts by the people had disappeared, there seemed to be a void at this moment which was filled by our present Offertory prayers (Thalhofer, op. cit. below, II, 161). For a long time these prayers were considered a private devotion of the priest, like the preparation at the foot of the altar. They are a Northern (late Gallican) addition, not part of the old Roman Rite, and were at first not written in missals. Micrologus says: "The Roman order appointed no prayer after the Offertory before the Secret" (cxi, P.L., CLI, 984). He mentions the later Offertory prayers as a "Gallican order" and says that they occur "not from any law but as an ecclesiastical custom". The medieval Offertory prayers vary considerably. They were established at Rome by the fourteenth century (Ordo Rom. XIV., 53, P.L. LXXVIII, 1165). The present Roman prayers were compiled from various sources, Gallican or Mozarabic. The prayer "Suscipe sancte pater" occurs in Charles the Bald's (875-877) prayer book; "Deus qui humanæ substantiæ" is modified from a Christmas Collect in the Gregorian Sacramentary (P.L., LXXVIII, 32): "Offerimus tibi Domine" and "Veni sanctificator" (fragment of an old Epiklesis, Hoppe, "Die Epiklesis", Schaffhausen, 1864, p. 272) are Mozarabic (P.L. LXXXV, 112). Before Pius V's Missal these prayers were often preceded by the title "Canon minor" or "Secretella" (as amplifications of the Secret). The Missal of Pius V (1570) printed them in the Ordinary. Since then the prayers that we know form part of the Roman Mass. The ideas expressed in them are obvious. Only it may be noted that two expressions: "hanc immaculatam hostiam" and "calicem salutaris" dramatically anticipate the moment of consecration, as does the Byzantine Cherubikon.
While the Offertory is made the people (choir) sing a verse (the Offertorium in the sense of a text to be sung) that forms part of the Proper of the Mass. No such chant is mentioned in "Apost. Const."; VIII, but it may no doubt be supposed as the reason why the celebrant there too prays silently. It is referred to by St. Augustine (Retract., II, xi, P.L., XXXII, 63). The Offertorium was once a whole psalm with an antiphon. By the time of the Gregorian Antiphonary the psalm has been reduced to a few verses only, which are always given in that book (e.g., P.L., LXXVIII, 641). So also the Second Roman Ordo: "Canitur offertorium cum versibus" (ib., 972). Durandus notes with disapproval that in his time the verses of the psalm are left out (Rationale, IV, 26). Now only the antiphon is sung, except at requiems. It is taken from the psalter, or other book of the Bible, or is often not a Biblical text. It refers in some way to the feast or occasion of the Mass, never to the offering of bread and wine. Only the requiem has preserved a longer offertory with one verse and the repetition of the last part of the antiphon (the text is not Biblical).
At high Mass, as soon as the celebrant has chanted the Oremus followed by no prayer, the choir sings the Offertory. When they have finished there remains an interval till the Preface which may (when the organ is permitted) be filled by music of the organ or at any time by singing some approved hymn or chant. Meanwhile the celebrant first says the Offertory chant. The corporal has been spread on the altar during the creed. The subdeacon brings the empty chalice and the paten with the bread from the credence table to the altar. The deacon hands the paten and bread to the celebrant. He takes it and holding it up says the prayer: "suscipe sancte Pater". At the end he makes a sign of the cross with the paten over the altar and slips the bread from it on to the corporal. Soon after the paten is given to the subdeacon's charge till it is wanted again for the fraction. The deacon pours wine into the chalice, the subdeacon water, which is first blessed by the celebrant with the form: "Deus qui humanæ substantiæ". The deacon hands the chalice to the celebrant, who, holding it up, says the prayer: "Offerimus tibi Domine". The deacon also lays his right hand on the foot of the chalice and says this prayer with the celebrant — a relic of the old idea that the chalice is in his care. The celebrant makes the sign of the cross with the chalice and stands it behind the bread on the corporal. The deacon covers it with the pall. The celebrant, bowing down, his hands joined and resting on the altar, says the prayer: "In spiritu humilitatis"; rising he says the "Veni sanctificator" making the sign of the cross over all the oblata at the word benedic. Then follows the incensing of the altar and the Lavabo. The use of incense at this point is medieval and not originally Roman (remnant of the incense at the Gallican procession of the oblata?). Micrologus notes that the Roman order uses incense at the Gospel, not at the Offertory; but he admits that in his time (eleventh century) the oblata are incensed by nearly everyone (De Exxl. Observ., IX). Finally, after the Lavabo the celebrant at the middle of the altar, looking up and then bowing down, says the prayer "Suscipe sancta Trinitas" which sums up the Offertory idea. The Orate fratres and secrets follow.
At low Mass, the parts of the deacon and subdeacon are taken partly by the server and partly by the celebrant himself. There is no incense. At requiems the water is not blessed, and the subdeacon does not hold the paten. The Dominicans still prepare the offering before Mass begins. This is one of their Gallican peculiarities and so goes back to the Eastern Proskomide. The Milanese and Mozarabic Missals have adopted the Roman Offertory. The accompanying chant is called Sacrificium at Toledo. http://www.newadvent.org/cathen/11217a.htm