Who were the three wise men?

Answer

In the original Greek, Matthew's Gospel originally referred to magoi (Latin: magi, sing: magus), not wise men. The magi were priests of the Zoroastrian religion. Perhaps Matthew wanted to show that even the priests of this great religion wanted to worship Jesus. Later, the Zoroastrian connection became less important and many translations now call them "wise men". Other modern interpretations are that they were kings. John Shelby Spong (A Bishop Rethinks the Birth of Jesus) says that among people he knows in New Testament circles, the universal assumption is that the magi, or wise men, were not actual people.
 

Answer

We do not know what their names are as the Bible does not say. The important thing is not who they were, but why they were there and why the Jewish scribes and Pharisees who should have known about the coming of the Messiah were not aware of the significance of Jesus's birth. (Incidentally, there were more than just 3 wise men: some scholars estimate there were up to one thousand people in their entourage from Persia.)

Medieval tradition says that their names were Caspar, Balthazar and Melchior, but it is uncertain where this information originated. It is also not certain what their role was, exactly. They are variously regarded as astrologers, magi and/or kings.