The three "kings" mentioned in John Henry Hopkins' Christmas carol are a reference to a story mentioned in the New Testament in Matthew 2:

Matthew 2:2

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?"

Matthew 2:11

On coming to the house, they saw the child with his mother Mary, and they bowed down and worshiped him. Then they opened their treasures and presented him with gifts of gold, frankincense and myrrh.

Note that it does not actually mention that there were three of them, or whether they actually were kings or not, nor where they came from nor their names.

However, in the Old Testament there are various prophecies that the verse in Matthew seems to fulfill

Isaiah 60:6:

The multitude of camels shall cover your land, The dromedaries of Midian and Ephah; All those from Sheba shall come; They shall bring gold and incense, And they shall proclaim the praises of the Lord.

Psalms 72:10

The kings of Tarshish and of the isles will bring presents; The kings of Sheba and Seba Will offer gifts.

If the story in Matthew is a fulfilment of the prophecies in Isiah and Psalms then we can assume the Magi were in fact kings and came variously from Midian, Ephah, Sheba, "Tarshish and the Isles" and Seba, Three "gifts" suggests there were three of them.

Note that the only source we have is the Bible itself (New King James Translation) and as far as I am aware there are no other sources that mention this event. To what extent we are dealing with myth or history is an open question.

The problem now lies with identifying exactly where these places are. This has been a matter of debate for centuries and, for some, there is no clear consensus:

• Midian - Believed to be northwest Arabian Peninsula, on the east shore of the Gulf of Aqaba on the Red Sea [Dever, W. G. (2006), Who Were the Early Israelites and Where Did They Come From?, William B. Eerdmans Publishing Co., p. 34, ISBN 978-0-8028-4416-3]
• Ephah - The tribe of Ephah is mentioned in inscriptions of the 8th-century BC Assyrian kings Tiglath-Pileser III and Sargon II, making it the only tribe of the Midianites to be attested outside of the Bible. Their homeland is thought to have been around Yathrib, in the Medina Province of Saudi Arabia. [Israel Ephʻal, The Ancient Arabs: Nomads on the Borders of the Fertile Crescent, 9th–5th Centuries B.C. (Magnes Press, 1982), pp. 216–217}
• Sheba - Probably somewhere in Southern Arabia [Israel Finkelstein, Neil Asher Silberman, David and Solomon: In Search of the Bible's Sacred Kings and the Roots of the Western Tradition, p. 167] but alternative theories put it in Sudan, Egypt, Ethiopia, Iran or Nigeria.
• Seba - This could be a confusion in the minds of the original authors with Sheba as the two terms were used to some extent interchangeably but if it existed as a separate kingdom then it may have been the southern Levant or even the ancient kingdom of Minoa which was spread across the Greek Islands including Corfu and Crete [Hommel, Südarabische Chrestomathie (Munich, 1892), p. 64.]
• Tarshish and the Isles - Legends have grown up around Tarshish over time and its identity has been the subject of scholarly research and commentary for more than two thousand years. It seems to have been a port and/or group of Islands in the Mediterranean. Likely candidates are Sardinia, Cadiz or Tartessos in Spain [Thompson, C.M.; Skaggs, S. (2013). "King Solomon's silver?: Southern Phoenician hacksilber hoards and the location of Tarshish'"] or even Britain [Smith, George (1856). Sacred Annals; Or, Researches into the History and Religion of Mankin[d]. p. 557]