Rebuilding the Foundations
As the story goes, Ptolemy, pharaoh of Egypt, wanted to build the world's greatest library in Alexandria. He heard about the books of the Hebrews, so he sent to the high priest Eleazar to get 70 Jewish scholars to translate the Hebrew Scriptures for his library. He put all 70 in separate rooms but, amazingly, they all produced exactly the same translation, word-for-word, without error. (Justin. Hortatory Address to the Greeks 13. c. AD 150.)
Only the most superstitious believe this actually happened. Justin was part of what could be called the "King James Only" movement in the early churches that believed the Septuagint was inspired.
The belief that the 70 Jewish elders really produced 70 exactly word-for-word translations was probably common in the early church, but it was by no means universal. By the early 200's, Origen would produce the "Hexapla," which had six different versions of the Old Testament, only one of which had the Septuagint in it. He also included a late second century of the translation from Hebrew to Greek by Theodotion. Since Origen was careful to mark the places in the Septuagint which did not match the Hebrew, it is clear that he did not consider the translation itself to be inspired word-for-word.
Despite the various attempts to update the Septuagint, the Greek-speaking eastern half of the Roman Empire continued to use the Septuagint as their Old Testament, and the Christians who descended from them—the various Orthodox churches—continue to use it to this day. The Orthodox Study Bible can be purchased at any bookstore, and its Old Testament is translated from the Greek Septuagint.
Most Christian writings from the western half of the Roman empire were written in Latin from the third century onward. As a result, Jerome produced a Latin translation made directly from the Hebrew. It became known as the Latin Vulgate (from "vulgar," meaning "in the people's language"), and though it was updated around AD 800, it became the official Bible in the west (the Roman Catholic Church) for the next thousand years.
Current Protestant and Roman Catholic Old Testaments lean on the "Masoretic text." That is an official Jewish texted preserved by the Jews since ancient times. The problem with the Masoretic text is that we have no Masoretic manuscripts dating any earlier than the ninth century.
The question, then, is which is more reliable: the Septuagint, for which he have texts hundreds of years older as well as thousands of quotes in the writings of the early Christians, or the Masoretic text, which is Hebrew and not a translation, but which may have changed over the centuries?
The question is whether centuries of copying or a session of translation loses more accuracy. The Septuagint is based on a much older Hebrew original, but it is a translation. The Masoretic is not a translation, but it had centuries of copying and copy errors since the Septuagint was translated.
Further complicating things is that it is the Jews who preserved the Masoretic text, not the Christians. The Jews were not fond of the Christians, nor of the way the Christians interpreted "their" Scriptures. At least one Christian, Justin Martyr of the mid-second century, accused the Jews of changing the Scriptures on purpose to hide the way they proved Jesus was the Messiah (Dialogue with Trypho 71.)
I've heard it said that all the Old Testament quotes in the New Testament are from the Septuagint. That's not true, but the majority are and probably the vast majority.
Catholic.com has a great page on this. Note that they quote a Protestant source for their answer, so this link is a place where Protestants and Catholics are agreed on a fact of history. I wouldn't even have quoted that page had they not had a good source.
Another fellow, with an incredible amount of time on his hands, took the time to give what he considers to be every Old Testament quote found in the New Testament, and he gives the English translations of both the Septuagint and Hebrew for the verse being quoted. As a result, you can look at every one and determine for yourself which Old Testament is most used by the apostles.
The Septuagint, as used in most eastern churches, has more books than either the Protestant or Catholic Bible. Read about why at The Deuterocanon/Apocrypha.
I count the page on the Gospel of the Kingdom to be the most important page on this site. Matthew Bryan, who wrote a book on the subject called Forgotten Gospel, has a great trio of articles beginning with this one on his blog/web site.