The priesthood is the eternal power and authority of God. Through the priesthood God created and governs the heavens and the earth. Through this power He redeems and exalts His children, bringing to pass “the immortality and eternal life of man” (Moses 1:39). God gives priesthood authority to worthy male members of the Church so they can act in His name for the salvation of His children. Priesthood holders can be authorized to preach the gospel, administer the ordinances of salvation, and govern the kingdom of God on the earth. — Priesthood, Gospel Topics, www.lds.org
In the Church today, worthy male members may receive the Aaronic priesthood ordinances and give service. As they worthily fulfill their duties, they act in the name of the Lord to help others receive the blessings of the gospel. — Aaronic Priesthood, Gospel Topics, www.lds.org.
The restoration of the Melchizedek Priesthood occurred some time later.
Through the authority of the Melchizedek Priesthood, Church leaders guide the Church and direct the preaching of the gospel throughout the world. In the ordinances of the Melchizedek Priesthood, “the power of godliness is manifest” (D&C 84:20). This greater priesthood was given to Adam and has been on the earth whenever the Lord has revealed His gospel. It was taken from the earth during the Great Apostasy, but it was restored in 1829, when the Apostles Peter, James, and John conferred it upon Joseph Smith and Oliver Cowdery. — Melchizedek Priesthood, Gospel Topics, www.lds.org.
Once the priesthood authority was restored, then baptisms, confirmations, baby blessings, administering of the Sacrament and all other ordinances were finally being performed with proper priesthood authority and priesthood keys.
The Church was now being administered by the authority of one holding the Holy Melchizedek Priesthood, and Joseph Smith, Jr. understood the magnitude of that authority, both what it represented and how he was required to behave in order to be worthy of that priesthood.
This I know. With the restoration of the priesthood came a direct authority bestowed on mankind by a kind and loving Savior. With that priesthood the lives of men, women and children are blessed beyond any one person’s richest imagination.