In the Gospel of John (19:19-20) the inscription is explained:
"Pilate also had an inscription written and put on the cross. It read, 'Jesus of Nazareth, the King of the Jews.' Many of the Jews read this inscription, because the place where Jesus was crucified was near the city; and it was written in Hebrew, in Latin, and in Greek."
Latin for "Jesus of Nazareth, the King of the Jews" is "Iesus Nazarenus Rex Iudaeorum" and the acronym for this title is INRI.
Pilate was loathe to crucify Jesus without some strong justification. The one he used was the standing Roman treaty with the Jews which allowed them self-government. When it was complained that Jesus was interfering with that self-rule by holding himself to possess authority that the lacked, Pilate challenged him to deny that he was called "King of Jews." Jesus did not deny the accusation. Hence, the justification for his crucifixion, that he claimed an illegitimate title, was made a part of the final punitive act.
Some Eastern Orthodox depictions of the Crucifix have a slight variation, showing instead the letters INBI, based on the Greek text of the inscription on the cross (Iésous o Nazóraios o Basileus tón Ioudaión), rather than on the Latin text.