I was explaining to a class what the issue is with regard to verses that are in the KJV but not in other versions. I decided to explain in story form:
In my version of the story, in the first three centuries of Christianity, the books of the New Testament were copied everywhere. They were not all in one book but on separate scrolls (the precise contents of the NT were not fully agreed on yet). Churches copied these scrolls as they were able to get hold of them from other churches. Most people, of course, were illiterate and so only heard the text read or had passages memorized. The vast majority of Christians would not have had personal copies of these scrolls. Perhaps not even many house churches would have copies of them.
There were no doubt many minor variations between all these scrolls, although not nearly the free-for-all that some might make us fear. When Christianity became legal in AD313, the next century turned to the standardization of many things. It was in the 300s that the Trinity was settled. Near the end of the 300s Christians finally agreed on what books belonged in the New Testament. And, I would argue, there was increasing standardization of the wording of the New Testament text.
This standardized text focused on clarity, both in style and in meaning. The text that resulted was a magnificent text, more or less the Greek text behind the King James Version (KJV). It is the text used by Christians everywhere for the next 1500 years and is still used by those who read the King James Version or its offshoots. I want to make it clear that I think it is perfectly acceptable to read the Bible from this text. If God was okay with it for 1500 years, it must be okay!
It includes passages like Mark 16:9-20, which most think was added to fill out Mark's ending. It includes the story of the woman caught in adultery in John 8, which many think may be a historical story even if it would prove not to have been in the original edition of John. It included a few other famous instances like Acts 8:37 and an extended 1 John 1:7.
It is no surprise that the majority of manuscripts read like this textual tradition. Most of the manuscripts we have date from the Middle Ages. We have only a handful of texts from the 300s and before. The discovery of much older copies of the NT than were available at the time of the KJV in the 1800s and 1900s prompted a re-examination of how the original texts might exactly have read.
So the earliest witnesses and internal evidence suggest that Mark 16:9-20 probably wasn't the original ending of Mark. But the verses mentioned above are the main differences. The overwhelming majority of the text is the same whether you use the KJV or a modern translation, almost all of which are based on the re-examination prompted in the 1700s and 1800s.
So that's my storied version of what I think happened and why the underlying Greek text is slightly different between the KJV and almost all the other versions.