Today is the feast of St. Matthew, Apostle and Evangelist. Most of what we know about him directly comes from his call:
As Jesus was walking along, he saw a man called Matthew sitting at the tax booth; and he said to him, “Follow me.” And he got up and followed him. And as he sat at dinner in the house, many tax collectors and sinners came and were sitting with him and his disciples. When the Pharisees saw this, they said to his disciples, “Why does your teacher eat with tax collectors and sinners?” But when he heard this, he said, “Those who are well have no need of a physician, but those who are sick. Go and learn what this means, ‘I desire mercy, not sacrifice.’ For I have come to call not the righteous but sinners.” (Mt 9:9-13 NRSV)In 10:3, he is again called a tax collector; other than that, he is just mentioned as one of the Twelve. In the very similar corresponding passages in Mark 2 and Luke 5, the tax collector is called Levi; Mark specifies further that he is the "son of Alphaeus". On the basis of this, Matthew and Levi are almost universally identified; because another one of the Twelve is also usually identified as 'James, son of Alphaeus', there is a common tradition that they were brothers. There are technically other possibilities consistent with the texts we have -- e.g., that Matthew, Levi, and James were three brothers -- but all of them are more complicated and none of them show up in the tradition. In the first century people using different names for different purposes was not uncommon; Judeans in Judea and Galilee had a much more fluid sense of names than we do. (One idea I once heard, which is pure speculation, but which has always stuck in my mind, is that Jesus, whom we know sometimes to give his disciples names, called Levi and gave him the name 'Matthew', Gift of the Lord, as a way of putting his past behind him.)
Some people have suggested that Matthew may also be the 'Mattai' who is one of the five disciples of Yeshu the Nazarene in Sanhedrin 43a; but it's hard to say. The rabbis may be conflating Jesus with another, earlier person of the same name, and in any case they are not interested in the history but in the question of whether something in a preserved baraita (an oral tradition not found in the Mishnah) conflicted with a given interpretation of the law. They effectively conclude that it doesn't because the Yeshu case was a weird exception due to the practical conditions. Of the five disciple names given, the others (Nakai, Netzer, Buni, and Toda) seem to be chosen just to make the point that the case was a unique one, which suggests that we should not put much weight on it. Nonetheless, we can't rule out that the original story was partly inspired by Matthew's name to begin with, and the play-on-words it allowed, so we don't know.
According to tradition, Matthew mostly spent his life preaching in Judea, but eventually went to preach to the Jewish community elsewhere -- most traditions and legends say Ethiopia, although by a weird quirk of naming, this could mean either Africa or modern-day Georgia, we don't know. He is traditionally regarded as a martyr, although we don't know how he died; one later legend suggests that he was killed by sword during liturgy.
His status as an Evangelist is due in part to the influence of Papias, who says in a quotation preserved by Eusebius that Matthew collected the logoi or sayings of Jesus in the Hebrew dialect, and others interpreted it. It's unclear whether the Gospel was attached to the name of Matthew because it seemed like the one that best fit the claim of Papias or whether Papias says it because it was already seen as Matthew's. In either case, it is a very reasonable conclusion that the sayings Papias mentions and the Gospel of Matthew are connected; parts of the Gospel as we have it are clearly arranged around sets of prophetic texts connected to Jesus' life and around sets of parables, so it's entirely possible to think of the Gospel as we have it as being such an original set of sayings 'interpreted'. (If you accept Markan priority, the 'interpretation' could very well have been in part by integrating it with Mark.)
What is interesting is that Matthew is consistently associated with writings specifically for Jewish Christians; this is true not only of Papias's tradition and the superscription for the Gospel, but also of the various heretical works attributed to Matthew. James son of Alphaeus is sometimes identified with James the Less, who was one of the Brothers of the Lord. The Brothers of the Lord were relatives of Jesus who played (as we see at the beginning of Acts and in 1 Corinthians 9) an important leadership role in the early Christian community, especially around Jerusalem, up to the destruction of the Temple. There are complications with identifying James son of Alphaeus and James the Brother of the Lord, but if they are identified, then it would mean that Matthew may have been one of the Brothers of the Lord too. (The brothers of the Lord are named in Mark 6 and Matthew 13 as James, Joseph, Simon, and Judas, but we don't know if the list is intended to be exhaustive or just representative.) I'm inclined not to make the identification between the two Jameses, but it's certainly one that has occasionally been made.