Description: An indeterminate number (but definitely more than one) of scholars/learned men who originate from a land far to the East of Bethlehem. They are likely to have possessed substantial wealth and some significant importance in their home lands. Astrology and astronomy were most likely some of their main focuses, but they also clearly knew a lot about history and at least some knowledge of Jewish tradition.
Significance: Their arrival tells us that the child they've come to visit is far from ordinary, and the gifts they bring with them tell us important facts about the boy they are being given to. Their inclusion in the incident makes an important point, that Jesus is not merely a figure of significance for Israel. His calling of figures as important as these tells us that he is a gift to the whole world.
First Sighting: They emerge only once, in Matthew 2 between verses 1-12, but it is very clear that their arrival was noteworthy.
"After Jesus was born in Bethlehem in Judea, during the time of King Herod, Magi from the east came to Jerusalem and asked, 'Where is the one who has been born king of the Jews? We saw his star when it rose and have come to worship him.' When King Herod heard this he was disturbed, and all Jerusalem with him."
(Matthew 2, Verse 1-3)
Their generosity is recorded later, as is their faithfulness to God's plan in protecting Jesus from Herod.
"On coming to the house, they saw the child with his mother Mary, and they bowed down and worshipped him. Then they opened their treasures and presented him with gifts of gold, frankincense and myrrh. And having been warned in a dream not to go back to Herod, they returned to their country by another route."
(Matthew 2, Verses 11-12)
Earlier prophecies about them are believed to be found in three separate occasions.
"Nations will come to your light, and kings to the brightness of your dawn."
(Isaiah 60, Verse 3)
"May the kings of Tarshish and of distant shores bring tribute to him. May the kings of Sheba and Seba present him gifts."
(Psalm 72, Verse 10)
"Because of your temple at Jerusalem, kings will bring you gifts."
(Psalm 68, Verse 29)
Explanation: The word Magi that we see in our Bibles today comes from the Greek word Magos. This in turn is derived from a language originating in modern day eastern Iran, Avestan. That language only remains known to this day because of its use in the Holy Texts of the Zoroastrian faith.
It is this that gives us our first clue as to who these people were. They were members of the Zoroastrian priestly caste. As the 'astria' part of their name suggests, they were concerned with constellations, stars and the movement of planets. Hence their extreme interest when they first sighted the star of Bethlehem. Their knowledge of astrology, and their understanding of the sacred texts of other cultures, told them this was something to investigate.
Although the songs say that there were three kings of orient, the truth is the Bible never specifies how many there were, nor that they were kings. We can infer however that they were individuals of substantial wealth and standing, as they managed to not only gather together the resources to go on an expedition, but also the money to buy very expensive gifts for a king that they only had their academic learning to fall back on to prove that he even existed. In order to gather this much money and command this much loyalty from everyone, from servants to security guards (the road they would be travelling on would have been notorious for banditry), they must have had some substantial degree of authority.
Kingship is perhaps the most obvious, but they could have also been deans or chancellors of royally funded universities or other scholarly institutions. They either possessed great influence among the powerful and wealthy, or it was they themselves who were in possession of great power and wealth.
In term of their numbers, we know that it was more than one, because of the plural often used, but beyond that we can only speculate. Later traditions of the three kings can be traced to two places. Firstly, there's the fact that there are three gifts, thus forever immortalising the giving of gifts at Christmas. Lots of people would have seemed to have inferred a "one gift per king" theory, which looking through the lens of contemporary culture, makes a level of sense (giving more than one gift in a Secret Santa kind of misses the point).
Secondly, there are traditional writings of many early church traditions that don't just tell us how many magi there were, but also their names and where they were from. Melchior, Caspar, and Balthazar are their names in the western tradition, and they were kings of provinces in Arabia, Persia, and India respectively. This originates from a document written in Greek, most likely in the city of Alexandria in around 500 AD, far later than anything else considered biblically canonical.
Exactly where they came from is unclear. While Matthew tells us they are "from the east" this doesn't narrow it down significantly. The traditional view has linked them to many different places, with one 14th century historian, John of Hildesheim, pointing out that there is a traditional story from the ancient silk road city of Taxila (near modern day Islamabad) that one of the Magi passed through there on their way to Bethlehem.
However, evidence of this, and of other details about them, is unclear. There are some modern traditions in China that suggest that one of the Magi hailed from their country. The Zoroastrian connection means that the strongest likely link is to the Persian region, or any other area with substantial Zoroastrian influence at the time. Beyond that however, we cannot be sure.