I will give you my understanding, just keep in mind I am no expert on textual criticism.
When speaking about the majority text, there are hundreds of manuscripts with small differences between them. Differences can be introduced by hand-copying which was the way the Bible text was propagated before the printing press was invented.
The way the text is determined is by comparing the manuscripts. The correct text is based on the agreement, since the disagreements are not in the same place in all manuscripts, there is a large body of agreement on any given text. What comes out of this is a text that is complete based on the body of agreement of the various texts.
Most of the modern versions are based on what is called the critical text, rather than the majority text. The critical text has many less copies to compare and precedence is given to the oldest (I think). This is based on archeological finds, not on transmission through the years.
Any way you look at it, the differences between the underlying text, whether majority text or critical text are not large, but for accuracy I would look for an english translation of the majority text.
Now we get into the problem of translation into other languages, such as english. Any time you translate into another language, there are choices made by the translators where there is not a strictly equivalent word for word translation. Also, phrasing is different in different languages. A direct word-for word translation of Hebrew might not be very readable in English.
English Bibles fall into rough categories (most in between the extremes) of a literal translation or a dynamic equivalence translation. The NASB is pretty literal, the NIV is pretty much dynamic equivalence, most others fall in between. This is the translation philosophy.
The differences in Bibles are generally because of the use of a different underlying text (MT or CT), or different choices by the translators. As Christ promised to preserve His word, we trust that He has guided the translation of the most reliable version of the english Bible. In any case, the argument is that no critical doctrines are affected by any of the differences.
There are plenty of underlying manuscripts to go to if you wanted to validate the deeper meaning of a word or phrase. To say we don't have an accurate Bible available today shows a lack of understanding of what has gone into producing them. The diffences are what people are focusing on when they say we don't have an accurate Bible today. Don't focus on differences, but on agreement. You will find the same ideas conveyed in any mainstream Bible.
