What does 'INRI' stand for on the crucifix?

It's Latin. It stands for "Iesus Nazarenus Rex Iudaeorum."

That means "Jesus of Nazareth, King of the Jews."

"Iesus" is the Latin word for Jesus, because the initial "J" is simply written as an "I." Latin doesn't use the letter "J." The word "Nazarenus" is, of course, the Latin genitive for the city where Jesus was from, Nazareth. Because of the ending on the word, the word "of" is understood. The word "Rex" is Latin for "king." And finally, "Iudaeorum" means "of Jews," with the initial "J" being replaced again by the Latin "I." The English word "Judaism" is derived from this same Latin word.

How do we know what it means? Because the Bible tells us so. In the Gospel according to John, Chapter 19, beginning at Verse 19, it reads:

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. Then the chief priests of the Jews said to Pilate, "Do not write, 'The King of the Jews,' but, 'This man said, I am King of the Jews.' " Pilate answered, "What I have written I have written." [John 19:19-21 NRSV]

OTHER RELIGIOUS USES OF THESE LETTERS

Those letters were also on the Jesuit flag, on which they stood for another, very different, Latin saying: "Iustum Necar Reges Impios" (It is just to annihilate heretical kings, governments, or rulers). The Jesuits are a religious order in the Roman Catholic church; they still exist today. Several priests are Jesuits. The order is known for fostering advanced education. In fact, many Jesuits received a doctorate in the sciences, the humanities, or the arts before they became priests.

On the crucifix, however, the letters are the Latin acronym (first letters of the words) for "Jesus of Nazareth, King of the Jews."