Christians say there were 12 apostles, but we can find more than twelve apostles mentioned in the Bible (like Paul & Barnabas) Where did these other apostles come from?
The office of an apostle is one Jesus instituted and only He can appoint those who hold the office. The term apostle means "one sent with a message," and the Lord is the One Who sends an apostle.
Clearly, Jesus appointed more than 12 men as apostles, as you pointed out. In fact, in the Bible there are two groups of apostles. First there are those who were called "the twelve." The twelve apostles stand apart from the rest of the apostles, since they will judge the twelve tribes of Israel:
Specifically, the twelve apostles are the eleven men Jesus chose while He walked the earth before His death (not counting Judas, who died), and Matthias, who the Lord chose after His acsension through the means of lots:
Notice in vs.20-21, this selection of this new man to replace Judas was proper and ordained by the Lord because it was according to prophecy, which foretold that a new man must be chosen to replace Judas. Matthias and Joseph were the only two men who could be considered for the position, because only these two men could pass the tests required to be one of the twelve: they had accompanied Jesus since His baptism by John and had seen the resurrected Lord in the flesh.
Apart from these twelve, there were other apostles as well, but all were appointed personally by the Lord. As you mentioned, Paul was an apostle, and he read about his personal appointement by Christ in the Acts. Likewise, Barnabas and James were apostles, and though we don't have record of their appointment in scripture, we can be sure they were appointed by Christ in the days after His resurrection.
Undoubtedly, there were other apostles unnamed in scripture as well, but regardless all true apostles met the test of having witnessed the resurrected Lord and proved their office by the unique gifts of an apostle, including raising men from the dead, healing supernaturally, surviving deadly snake bites, etc.
Today, apostles no longer exist in the church, because there are no men remaining alive who can meet the test of seeing the resurrected Lord.