How accurate is the text in the Bible?

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Ancient biblical manuscripts, like other ancient literature, provide a lot of variant readings, though most are pretty minor and don't change the meaning much. There is a lot of scholarship devoted to studying this and biblical translators rely on these findings, though what decisions they make about which version is most original or authentic can vary. You will see in many bibles that Mark has alternate endings that are demarcated from the rest of the book as being later additions. 

As to the reasons for these variations: some can be typos, others could be attempts to 'fix' things which the editor believed were typos which may or may not have actually been. Some edited or added to the text because they thought it needed clarification. Sometimes texts are combined or rearranged. The story of the Woman Caught in Adultery was probably not originally a part of the Gospel of John, but was a passage that came from another source and was added to John because it was considered important for inclusion and that seemed like a good place to attach it. This textual history doesn't necessarily mean that it doesn't belong there.

The idea that the Bible we have is the result of any kind of concerted effort to change the text for ideological or political reasons doesn't seem plausible to me. Maybe some people tried that in one place or another but there was no real centralized control over textual transmission from any early church hierarchy.