The first and fundamental aspect of accuracy is experts’ shared consensus on the best available texts in the original languages of Hebrew and Greek that underlie all translations, based upon a far richer body of surviving manuscripts than we have for other ancient writings. Here’s a somewhat over-simplified rundown.
Hebrew versions of the Jewish Bible or Tanakh, which Christians call the Old Testament, employ Judaism’s authoritative Masoretic Text. The ancient Hebrew language did not use written vowels, so medieval scholars known as Masoretes added “vowel points,” markings that standardized oral traditions on the correct readings of the words. This led to the 9th Century Aleppo Codex, corroborated by the 11th Century Leningrad Codex and other rabbinical manuscripts.
These carefully preserved texts are virtually identical in wording, though with some variations in pointing, and provide the basis for Jewish and Christian Bibles. The 15th Century emergence of printed Bibles further standardized the Hebrew text. The 20th Century rediscovery of biblical books among the 2,000-year-old ”Dead Sea Scrolls” added certain variations that translators consider.
With the New Testament, thousands of Greek manuscripts and fragments from Christianity’s early centuries exist. Today’s translators work from a standard “eclectic” Greek version formulated from these by specialists in “textual criticism” who decide which variants are closest to the original writings. Technical note: “earliest” does not necessarily mean “best” manuscript.
The resulting shared resource for translations is the German Bible Society’s continually updated text with all important variations, most recently the 2017 “Nestle-Aland Greek New Testament, 28th Edition, with Critical Apparatus.” Major translations undergo periodic tweaking based on this ongoing work. A prominent evangelical, the late Bruce Metzger of Princeton Theological Seminary, was the American leader in this international project.
Modern English Bibles provide candid footnotes that alert readers to important textual variants, which rarely affect basic biblical doctrines.
One example, which may surprise some readers, is Jesus’ cherished words at the Crucifixion, “Father, forgive them for they know not what they do” (Luke 23:34). Evangelicals’ popular New International Version (NIV, 1978, updated 2011), admits in a footnote that “some early manuscripts do not have this sentence.” U.S. Catholicism’s official New American Bible (NAB, 1970, revised 2011, with Old Testament books Jewish and Protestant Bibles omit) explains in more detail that this sentence “does not occur in the oldest papyrus manuscript of Luke and in other early Greek manuscripts and ancient versions of wide geographical distribution.”
There’s similar textual ambiguity with the equally beloved passage about Jesus forgiving the adulterous woman in John 7:53 – 8:11.
Aspect number two is debates over the accuracy of the English translations drawn from those reliable agreed texts. It’s impossible for The Religion Guy to even summarize here all the renditions on the market, but readers can compare for themselves searchable full texts of 54 options at Bible Gateway. Also you’ll find basic info about many English editions at GotQuestions.org, on the Bible versions page.