An Apostle in the New Testament meaning of the word is one who carries Christ's words on His behalf. They are Christ's representatives. In Acts chapter 1, when the disciples were gathered to replace Judas Iscariot in his 'bishopric' with another man, these qualifications were given:
For it is written in the Book of Psalms, 'May his camp become desolate, and let there be no one to dwell in it'; and 'Let another take his office.' So one of the men who have accompanied us during all the time that the Lord Jesus went in and out among us, beginning from the baptism of John until the day when he was taken up from us--one of these men must become with us a witness to his resurrection. (Act 1:20-22)
So an Apostle is one who had seen the risen Christ and could be a witness to His resurrection.
Paul identifies himself as an Apostle in most of his letters. This is because the risen Christ had appeared to him on the road to Damascus, and had commissioned him to preach the Gospel to the Gentiles.
In the book of Galatians, Paul devotes the first two chapters to defending his apostleship.
Read the following verses and note Paul's argument for his own Apostleship to the Galatians:
Paul, an apostle--not from men nor through man, but through Jesus Christ and God the Father, who raised him from the dead-- (Gal 1:1)As an Apostle, Paul was a spokesperson for Christ. His words and teachings are to be regarded just as though it were Christ Himself speaking and teaching! This is the reason we can say that all Scripture is given by inspiration from God, because Christ entrusted the Gospel to His Apostles, and they in turn have committed it to pen and paper.
For I would have you know, brothers, that the gospel that was preached by me is not man's gospel. For I did not receive it from any man, nor was I taught it, but I received it through a revelation of Jesus Christ. (Gal 1:11-12)
But when he who had set me apart before I was born, and who called me by his grace, was pleased to reveal his Son to me, in order that I might preach him among the Gentiles, I did not immediately consult with anyone; (Gal 1:15-16)
And from those who seemed to be influential (what they were makes no difference to me; God shows no partiality)--those, I say, who seemed influential added nothing to me. On the contrary, when they saw that I had been entrusted with the gospel to the uncircumcised, just as Peter had been entrusted with the gospel to the circumcised (for he who worked through Peter for his apostolic ministry to the circumcised worked also through me for mine to the Gentiles), and when James and Cephas and John, who seemed to be pillars, perceived the grace that was given to me, they gave the right hand of fellowship to Barnabas and me, that we should go to the Gentiles and they to the circumcised. (Gal 2:6-9)
One of my favorite quotes comes from the Early Church Father, Irenaus. Irenaus wrote;
We have learned the plan of our salvation from no one else other than from those (Apostles) through whom the gospel has come down to us. For they did at one time proclaim the gospel in public. And, at a later period, by the will of God, they handed the gospel down to us in the Scriptures-to be the 'ground and pillar of our faith'.