Answer: the Papal headwear that you are referring to is called a Mitre
the three crowns represent the Popes three main roles. The first is as the Bishop of Rome secondly as the successor of St Peter and lastly as head of the universal church.
There are anti-Catholic conspiracy theorists that read meaning in to the Tiara. It should be pointed out though, that it is not a liturgical vestment and is not used by the current Pope.
The last Pope to don this headdress was Paul VI 30 June 1963. This was a special modern crown that was a gift to him from his previous archdiocese, Milan. Its history has been linked to the Jewish high priests Mitre as recorded by Josephus as a golden crown with 3 rows.
Its use began in the 4th century in a simpler form and became a stiff headdress in the 10th century, gradually taking its current form of 3 crowns. It is still used in heraldry accompanying the keys of St Peter beneath it. It is found on the Vatican city-state flag.

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