The Last Rites are officially called the Viaticum, the sacrament of the dying. In Roman Catholicism, the Last Rites consist of a priest administering the Eucharist to a dying person. A lay person who has been designated an Eucharist Minister can also offer the Viaticum.
According to the Church's Canon Law,
Can. 921 §1. The Christian faithful who are in danger of death from any cause are to be nourished by holy communion in the form of Viaticum.
§2. Even if they have been nourished by holy communion on the same day, however, those in danger of death are strongly urged to receive communion again.
§3. While the danger of death lasts, it is recommended that holy communion be administered often, but on separate days.
The rite begins with a confession or an act of contrition.
The dying person then repeats their baptismal promises or says the Apostles Creed.
The priest or eucharistic minister recites the Our Father prayer.
When the dying person is ready, communion is administered. (Remember that even a very small particle of a host is sufficient.)
The priest says: "This is the Lamb of God who takes away the sins of the world. Happy are those who are called to his supper."
The sick person responds: "Lord I am not worthy to receive you, but only say the word and I shall be healed."
The priest says: "The Body of Christ."
The sick person responds: "Amen."
After the sick person receives communion, the priest says: "May the Lord Jesus protect you and lead you to eternal life."
The rite concludes with prayer and blessing.
A complete rundown of the prayers and blessings said during the Last Rites can be found in the Catholic Encyclopedia entry on the Viaticum.