What Hebrew and Greek text is being translated?

For the Old Testament, modern translations generally work from the Masoretic Hebrew tradition, while also consulting the Dead Sea Scrolls, Septuagint, Samaritan Pentateuch, Syriac, and other ancient witnesses where relevant.

For the New Testament, modern translations such as the NIV and ESV use a critical Greek text constructed by comparing thousands of surviving manuscripts and other ancient witnesses.

That's different from the KJV, whose New Testament was based primarily on the Greek textual tradition available to its translators in the early 1600s, especially the Textus Receptus.

This explains why you'll sometimes see:

KJV: a verse or phrase appears in the main text
NIV/ESV: the phrase is absent, footnoted, or marked as a textual variant

That does not automatically mean the modern translators removed Scripture. It can mean that subsequent manuscript discoveries provided additional evidence about what the earliest text probably contained.

How faithfully did the translators translate that text?

This is a different question.

Even if two translations use essentially the same underlying Hebrew or Greek text, they can render it differently because translation isn't a mechanical word-substitution exercise.

For example:

Formal/essentially literal approach:

Try to preserve words, grammatical relationships, and structure as much as good English permits.

This is closer to the philosophy of the ESV and NASB.

Meaning-oriented approach:

Communicate what the original author meant in natural contemporary English.

This is more characteristic of the NIV.

Neither philosophy automatically equals “faithful” or “unfaithful.” A highly literal rendering can actually obscure meaning if the original expression is idiomatic.

The really important distinction

When someone says:

“The NIV changed God's Word!”

I'd ask:

“Which Greek or Hebrew reading are you saying the NIV changed, and what manuscript evidence supports your preferred reading?”

That's a much more productive question than simply comparing the NIV with the KJV.

For example, suppose the KJV has:

a particular phrase

while the NIV has:

a different phrase

We need to determine:

A. Do the underlying Greek manuscripts actually differ?
B. If they differ, which reading has the stronger textual evidence?
C. If the Greek is essentially the same, why did the translators render it differently?
D. Does the difference affect doctrine, or merely wording?

Those four questions can resolve a surprising amount of the “NIV is corrupted” controversy.

And there's an important point here: the NIV and ESV are not simply “different Bibles.” They frequently translate the same underlying Hebrew and Greek text. Their differences often arise from translation philosophy, English style, interpretation, and decisions concerning textual variants, rather than from one committee believing in Christianity and another rejecting it.