ANSWER: The book of Acts records the calling and conversion of Saul of Tarsus, a zealous persecutor of those who believed Jesus was the Messiah. Paul was personally chosen by the risen and glorified Christ to be His apostle to the Gentiles. He makes in clear in the book of Galatians that his special calling was "not from men nor through man" (Galatians 1:1), meaning no human or church group chose or 'ordained' Paul to be a minister or told him to preach to the Gentiles.
Paul's commission, which came directly from God, was known the day he was baptized (Acts 9:15-16). It did not take him long after his baptism to begin powerfully proclaiming that Jesus (who, only a short while before, he was persecuting - Acts 9:4) was the Son of God (see Acts 9:19 - 25).
Paul reiterates his direct calling and commission by stating, again in Galatians, that after his conversion he had no communication whatsoever with the other apostles (James, Peter, John, etc.) until many years later (Galatians 1:16).
The word apostle means "one sent." The twelve who had been with Jesus throughout His ministry were called to be eyewitnesses of His resurrection (Acts 1:21 - 22). When Judas Iscariot betrayed Jesus, he lost his part in this calling. Knowing that one of Jesus' other disciples needed to take Judas' part, the eleven found two among the 120 disciples who had also been with Jesus during His entire ministry and were present at His ascension - Matthias and Barsabas. Matthias was chosen by the risen Christ to take part with the eleven as an eyewitness of His resurrection (Acts 1:24 - 26).
Paul also became an eyewitness "as of one born out of due time" (1Corinthians 15:18). He personally saw the risen Christ and was taught by Him for 3½ years in Arabia (Galatians 1:17, 1Corinthians 11:2, 23) - which occurred after his conversion in Damascus.
The New Testament clearly establishes three criteria for the designation of apostle.
Seeing the risen Christ in order to be an eyewitness of His resurrection (Acts 1:22)
Being personally taught by Him for 3½ years (Acts 1:22)
Receiving a specific commission from Christ Himself (Acts 1:17, 25)
Paul fulfilled all three criteria and is therefore, without a doubt, an apostle. In closing, it should be noted that NO MAN, since the first century A.D., has met these three criteria for the designation of apostle.