Who leads the celebration of the Eucharist?
Actually Christ himself acts in every celebration of the Eucharist. The bishop or the priest represents him.
It is the Church's belief that the celebrant stands at the altar in persona Christi capitis (Latin = in the person of Christ, the Head). This means that priests do not merely act in Christ's place or at his command; rather, on the basis of their ordination, Christ himself, as Head of the Church, acts through them.
In what way is Christ there when the Eucharist is celebrated?
Christ is mysteriously but really present in the sacrament of the Eucharist. As often as the Church fulfills Jesus' command, "Do this in remembrance of me" (1 Cor 11:24), breaks the bread and offers the chalice, the same thing takes place today that happened then: Christ truly gives himself for us, and we truly gain a share in him. The unique and unrepeatable sacrifice of Christ on the Cross is made present on the altar; the work of our redemption is accomplished. (YOUCAT questions 215 & 216)
Dig Deeper: CCC section (1362-1367) and other references here.
Part 2: The Celebration of the Christian Mystery (1066 - 1690)
Section 2: The Seven Sacraments of the Church (1210 - 1690)
Chapter 1: The Sacraments of Christian Initiation (1212 - 1419)
Article 3: The Sacrament of the Eucharist (1322 - 1419)
V. THE SACRAMENTAL SACRIFICE: THANKSGIVING, MEMORIAL, PRESENCE ⇡
The sacrificial memorial of Christ and of his Body, the Church ⇡
The Eucharist is the memorial of Christ's Passover, the making present and the sacramental offering of his unique sacrifice, in the liturgy of the Church which is his Body. In all the Eucharistic Prayers we find after the words of institution a prayer called the anamnesis or memorial.
In the sense of Sacred Scripture the memorial is not merely the recollection of past events but the proclamation of the mighty works wrought by God for men.184 In the liturgical celebration of these events, they become in a certain way present and real. This is how Israel understands its liberation from Egypt: every time Passover is celebrated, the Exodus events are made present to the memory of believers so that they may conform their lives to them.
184.
Cf. Ex 13:3.
In the New Testament, the memorial takes on new meaning. When the Church celebrates the Eucharist, she commemorates Christ's Passover, and it is made present the sacrifice Christ offered once for all on the cross remains ever present.185 "As often as the sacrifice of the Cross by which 'Christ our Pasch has been sacrificed' is celebrated on the altar, the work of our redemption is carried out."186
185.
Cf. Heb 7:25-27.
186.
LG 3; cf. 1 Cor 5:7.
Because it is the memorial of Christ's Passover, the Eucharist is also a sacrifice. The sacrificial character of the Eucharist is manifested in the very words of institution: "This is my body which is given for you" and "This cup which is poured out for you is the New Covenant in my blood."187 In the Eucharist Christ gives us the very body which he gave up for us on the cross, the very blood which he "poured out for many for the forgiveness of sins."188
187.
188.
The Eucharist is thus a sacrifice because it re-presents (makes present) the sacrifice of the cross, because it is its memorial and because it applies its fruit: [Christ], our Lord and God, was once and for all to offer himself to God the Father by his death on the altar of the cross, to accomplish there an everlasting redemption. But because his priesthood was not to end with his death, at the Last Supper "on the night when he was betrayed," [he wanted] to leave to his beloved spouse the Church a visible sacrifice (as the nature of man demands) by which the bloody sacrifice which he was to accomplish once for all on the cross would be re-presented, its memory perpetuated until the end of the world, and its salutary power be applied to the forgiveness of the sins we daily commit.189
189.
Council of Trent (1562): DS 1740; cf. 1 Cor 11:23; Heb 7:24, 27.
The sacrifice of Christ and the sacrifice of the Eucharist are one single sacrifice: "The victim is one and the same: the same now offers through the ministry of priests, who then offered himself on the cross; only the manner of offering is different." "And since in this divine sacrifice which is celebrated in the Mass, the same Christ who offered himself once in a bloody manner on the altar of the cross is contained and is offered in an unbloody manner... this sacrifice is truly propitiatory."190
190.
Council of Trent (1562) Doctrina de ss. Missae sacrificio, c. 2: DS 1743; cf. Heb 9:14,27.