Why is INRI written on a cross and what is its significance?

What is the meaning of INRI on the cross above Jesus' head

The letters “INRI” are initials for the Latin title that Pontius Pilate had written over the head of Jesus Christ on the cross (John 19:19). Latin was the official language of the Roman Empire.

The words were "Iesvs Nazarenvs Rex Ivdaeorvm." Latin uses “I” instead of the English “J”, and “V” instead of “U” (i.e., Jesus Nazarenus Rex Judaeorum). 

The English translation is "Jesus of Nazareth, the King of the Jews."

INRI stems from the Latin phrase 'Iesus Nazarenus Rex Iudaeorum' meaning 'Jesus of Nazareth, King of the Jews'. This was the notice Pontius Pilate nailed over Jesus as he lay dying on the cross. It now signifies that a true Christian lies there.

Although John’s gospel refers to the writing as a “title,” Mark and Matthew both refer to it as an “accusation.” It was customary to set up over the heads of persons crucified the crime for which they suffered, and the name of the sufferer.

The accusation on which Jesus had been condemned by Pilate was his claiming to be the King of the Jews. Ironically, the “crime” for which Jesus was crucified is not a crime at all, but an absolutely true statement. Not only is Jesus King of the Jews, He is the King of all – the King of kings and the Lord of lords (Revelation 17:14 and 19:16). He is King over all the universe and all its inhabitants. And it was not any crime of His own that was nailed to the cross; it was the crimes (sins) of everyone who would ever put his or her faith in Him for salvation. He has “blotted out the handwriting of ordinances that was against us, which was contrary to us, and has taken it out of the way, nailing it to the cross” (Colossians 2:14).

Therefore many of the Jews read this title, because the place where Jesus was impaled was near the city; and it was written in Hebrew, in Latin, in Greek. However, the chief priests of the Jews began to say to Pilate: "Do not write 'The King of the Jews,' but that he said, 'I am King of the Jews.'" Pilate answered: "What I have written I have written."

Latin became the Language of the Roman Catholic Church. But, in Latin, they used the "I" for the Letter "J" (both in Jesus and Jews), their word for King was "REX", and they used the "V" for the "U" . So the sign would have been something like this one

IESVS NAZEREVS, REX IUDEA, Or using the first letters of each words: INRI

The final language on the sign was Hebrew which is the Jewish language. Now the first word was Yehoshuah (Jesus). The first letter of that is the smallest letter in the Hebrew alphabet. It is the little yod, which looks like an apostrophe. (Hebrew is read from right to left.) Yehoshuahand according to the Bible it was "Jesus the Nazarene." The next word was ha Nazarite. The next word would be "and," which is what we call a wha. It is like a little line with a hook on top. The next one would be the king of Jews Hamelech Yudio. What you had on the cross reading from right to left in the first letter, are the four letters of the Tetragrammaton YHWH. YHWH is the covenant name of God in the Old Testament Hebrew which did not have vowels (today we know this as Yahweh). 

In John 19:19-22 the chief Priests wanted Pilate to take the sign down because the sign recognized Jesus as Yahweh or God.

Just as the title King of the Jews was written in three languages, so do those of all nations and languages recognize Him as Savior, as indeed He is of all the elect of God whom He saves from all their sins, by bearing them in His own body on the cross, and of whom He is the able and willing, the perfect and complete, the only and everlasting Savior.