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.