This is a revised post from 2020, for the Feast of St. Matthias tomorrow.
Today is the feast of St. Matthias, Apostle. The story about him in Acts 1 is interesting in a number of ways. It occurs between the Ascension and Pentecost; Jesus has given his disciples their mission but they have not yet received the full measure of the Holy Spirit. Because of this, it often gets skipped over. But we learn a number of things from it. The disciples are meeting regularly in fairly large groups. The eleven Apostles left are explicitly mentioned, as are Mary the Mother of Jesus, the women, and the brothers of the Lord. (The women are mentioned not as if they were just a generic bunch of women but as if they were a well-defined and perhaps even formally defined group; they seem clearly to be part of the leadership. This fits with a number of things said in the Gospel of Luke, e.g., Luke 8:1-3, Luke 23:54-56; cp. Mark 15:40-41.) The gathering that chooses Matthias has about 120 disciples all told (which number may have only included the men, since Peter may have only addressed the men).
Peter is quite clearly the leader here; he tells them that Scripture says that Judas needs to be replaced (the word he uses is dei, i.e., 'It is required') and they do it. In fact, while it is never explicitly said, the whole thing is structured as if Peter had called the meeting specifically in order to do what they end up doing. Peter's reason is based on Scripture; he quotes Psalm 69 and Psalm 109. The latter is straightforward in its application ("May another take his place of leadership"), although the word for 'place of leadership' is actually 'supervision', episkopen. The other one reads a bit oddly in English: "May his place be deserted; let there be no one dwelling in it." It seems a little odd to quote that no one should dwell in his place in an argument that you should fill his place. But read in context, the verses both come from very similar passages: they are from the psalms that tend to embarrass people today, the ones in which the enemies of the psalmist are cursed. The verses in Acts 1:18-19, about what happened to Judas, are often read as parenthetical, but the thought of Peter's argument follows directly from them, not from Acts 1:17. The line of thought is: The Scripture had to be fulfilled which spoke of Judas (v. 16); Judas was one of their ministry (diakonias) (v. 17); with the payment for his injustice (adikias), he bought a field and died (v. 18); everybody in Jerusalem heard about it so called it the Field of Blood (v. 19); because Scripture says, "May his place be deserted...." and "May another take his place...." Thus Peter is reading the cursing passages of the Psalms as being about Judas. What it says about Judas in Psalm 69 is fulfilled by his death; so what it says about him in Psalm 109 must be fulfilled as well.
I find it interesting that they don't replace him until he is dead; the word for 'dwell' here (katoikon) suggests permanent settlement, so the curse on Judas is for his apostleship not to be permanent. This at least suggests very strongly, I think, why the Twelve did not keep replacing themselves as they died; they seem to have regarded the position as something distinctly attaching to each, each permanently dwelling it. (And this would fit with Jesus' comments about the twelve thrones of judgment, for instance.) Thus Judas cannot be replaced except under divine authority. The quotations are not rhetorical decorations, in other words; they are divine warrant for an action that Peter thinks would normally not be allowable. Acts shows us other people with apostolic ministry; but none of them, not even Paul or Barnabas, ever becomes one of the Twelve.
In any case, what Peter says is necessary is to make "one of these", i.e., the Twelve, from the men who accompanied the Lord Jesus the whole time from his Baptism to his Ascension. (This is interesting for indicating what Peter thinks of the Twelve, namely, that one of their major functions is specifically to be familiar with Christ's ministry so as to witness properly to the Resurrection. It also indicates why Luke begins by retelling the Ascension; it establishes the link to what immediately follows.) This in fact ends up being the entire backstory we know about Matthias: he was with Jesus the whole time from the Baptism to the Ascension. We know nothing else about who he was. The men in the assembly pick two -- Joseph Barsabbas, also called Justus, and Matthias. Later in Acts there is a figure named Judas Barsabbas, who is probably Joseph's brother; later tradition suggests Joseph was one of the Seventy in Luke 10 and afterward became bishop of Eleutheropolis, but as with Matthias, all we certainly know is that he was with Jesus the whole time from the Baptism to the Ascension.
But two is not "one of these". So what they do then is pray to God, knower of the hearts of all, that He will point out which one of the these two that He has chosen to take the place for this service (diakonias) and apostleship (aposteles) from which Judas traveled (the word could also mean 'die') "to his own place" (which obviously is an allusion to the 'place' mentioned in the Psalms). Then they cast lots. Casting lots was, of course, common for making decisions, as it is even now. Perhaps more likely on general grounds, lots were the standard way in which Temple duties were assigned. It is also possible, given the comment about Judas going to his own place, that they had Leviticus 16:8 in the background. In the atonement offering, the high priest makes an atonement before the Lord with the sacrifice of a bull and two goats. The goats are split, one for the Lord and one "for azazel" (in the Hebrew; we don't know for sure what the word meant) or "sent away" (in the Septuagint), by lot, and the one "for azazel" is then sent into the wilderness. Regardless, when the lots were cast, Matthias became one of the Twelve Apostles.
And that's the last we hear about him in Scripture. According to the most popular tradition, after preaching in Jerusalem a while he went down into "Ethiopia" (by which is likely not meant Ethiopia but Colchis in the Caucasus, in modern-day Georgia; Herodotus claimed that the Colchians were descended from the Ethiopians). For his death, the traditions are all over the place; he was martyred by crucifixion in Sebastopolis (in modern-day Turkey) or by stoning and beheading in Jerusalem or by stoning in Colchis, or he simply died of old age in Jerusalem.