Paul, an apostle of Jesus Christ by the will of God, and Timothy our brother, unto the church of God which is at Corinth, with all the saints which are in all Achaia: Paul, an apostle - Paul, commissioned immediately by Jesus Christ himself, according to the will of God, to preach the Gospel to the Gentiles. See on 1 Corinthians 1:1 (note).
In all Achaia - The whole of the Peloponnesus, or that country separated from the main land by the Isthmus of Corinth. From this we may learn that this epistle was not only sent to the Church at Corinth, but to all the Churches in that country.