Who was the 12th Apostle - Matthias or Paul? First off, a distinction between disciple and apostle - a disciple is a student, and the 12 disciples were Jesus' students.  An Apostle is one who was "sent out." Eleven of Jesus' disciples were sent out to preach the good news. Judas, of course, had committed suicide before being sent out.

The Disciples, recognizing the "need" for a 12th, cast lots and chose Matthias in Acts 1.  On the choice of Matthais, I've heard things similar to this:


  Also, we wonder if Luke was pointing out that casting lots, used all through the Old Testament, was a poor substitute for the guidance of the Holy Spirit, who became the source of wisdom and discernment for such decisions after Pentecost. 


As Lloyd Ogilive says:


  What happened to Matthias? We are not told. He never is mentioned again. Did he defect or drop out? Probably not. I believe we would have been told if that happened, and Luke’s thoroughness would have included that data. What we do know is that the position was filled by Paul. There is no need to be down on Matthias. He responded to a call. He was ready with his knowledge of Christ and an open mind and heart to receive His Spirit. He was there at Pentecost—that’s all that matters. Whether his ministry afterward received the recognition of history is unimportant. The same is true for us. Once we have experienced what Christ said and did for us in His death and Resurrection and then returned to continue to do, titles, or history’s recognition, or even the accolades of people today become unimportant.


The question is thus this - Did the Holy Spirit actually intend for Paul to be the real 12th Apostle, or was Matthias not just an accidental, rash act on the part of the Eleven?

Or, put more poetically (as well suggested by Flimzy, when it comes to apostleship, should we put our trust in God or men?)
There is one problem with the argument - "Matthias was never sent by Christ to do anything, therefore he cannot be an apostle". The problem is that Barnabus (of whom we have no evidence that he was sent by Christ) is also called an "apostle" in Acts 14:14. 

We know that Judas was an apostle and his "office" (Acts 1:20) was taken over by Matthias. And it is not as though this was done through some rash thinking on the part of the Peter, for he, along with the other apostles had in fact already received the Holy Spirit before Pentecost (John 20:22). 

Moreover, with the inclusion of Matthias, the phrase "Peter and the eleven" is used in Acts 2:14, which strongly suggests that Matthias was indeed the 12th apostle.