This is a subject I've become somewhat interested in recently, and the issue has turned out to be a lot more complex and nuanced than I initially realised. Broadly speaking, the Christian Bible can be broken into three parts with their own distinct history, and all three have generated controversy to a greater or lesser degree, either in their translation or in their status as scripture.
Parts of the Bible
The oldest is the Hebrew Bible, which is the collection of ancient Jewish writings that have survived in the original Hebrew (or, for a few passages, Aramaic), and which are considered canonical by Jews and all Christian denominations. These were first translated into Greek (a translation now known as the Septuagint) in the 3rd century BC, which is the source of most controversy surrounding the translation. For example, in Isaiah 7:14, the Hebrew word הָעַלְמָה ("young woman") was translated with the Greek παρθένος, which can mean either "maiden" or "virgin". Because Isaiah 7:14 is talking about a pregnancy, it would have been considered unlikely to be read as "virgin" at the time of its translation, but in a modern Christian context it is often taken as a prophecy of the birth of Jesus. Jews, who regard the original Hebrew text as authoritative, do not accept this Christian interpretation.
Next come the deuterocanonical books or apocrypha, terms preferred by Catholics and Protestants, respectively. These are pre-Christian Jewish writings which appear in the Greek Bible, but not the Hebrew Bible. These have widely varying histories. Some were originally written in Hebrew, but the original texts have been lost. Some are parts added to books in the Hebrew Bible — for example, around half of the Greek book of Daniel is nowhere to be found in Hebrew. Others, like 2 Maccabees, were entirely authored in Greek by Hellenised Jews. Whatever the reason for their absence from the Hebrew Bible, they are not considered canonical by Jews or Protestants, some are included in the Catholic Old Testament, and a few more are also included by Eastern churches.
Finally, there is the New Testament, thought to be originally written in Greek, although some proposals of a Hebrew original for the Gospel have been floated. This is, obviously, rejected by Jews, but it is universally accepted, as the same set of books, by all major modern Christian denominations. It is nevertheless controversial in the choice of manuscript used as the translation source. The most widely known English translation of the Bible, the King James Version (KJV), was created centuries ago based on the Byzantine manuscripts known at the time. Modern Biblical scholarship prefers the older Alexandrian text-type, which is missing some verses that were presumably added later to the Byzantine text-type. Therefore, from the point of view of someone familiar with the KJV, modern translations of the New Testament have "missing" verses.
Ancient translations
The issue of which source to use for translation is further complicated by the fact that no originals survive — and, indeed, for the older books of the Hebrew Bible, the concept of an original may be inapplicable, as these very likely originated as oral traditions that were only written down centuries later.
For the Old Testament, the main sources are the 10th-century Hebrew Masoretic Text (MT) and the 3rd-century-BC translation into Greek, the Septuagint (LXX), of which near-complete manuscripts survive from the 4th century AD. The Dead Sea Scrolls, from the 4th century BC, were discovered in the mid-20th century and have helped to improve the historical accuracy of recent translations.
The Latin Vulgate is also a useful point of reference, for two reasons. First, Jerome used the work of Symmachus when translating the Vulgate. Symmachus translated the Hebrew Bible into Greek, independently of the Septuagint. His translation is reported by ancient authors to be more natural Greek than the Septuagint, but survives only in fragments, making the Vulgate our best insight into Symmachus's work. Second, the Vulgate was the only version of the Bible ever read by most scholars in western Europe for about a millennium, thanks to the influence of the Roman Catholic Church.
For these reasons, modern Bible translations typically use multiple sources in different languages, especially for the Old Testament. Only by comparing different surviving manuscripts can we obtain a clear picture of what was meant by the original. Yet, for the selfsame reason, modern translations often differ in their reading of particular passages due to preferencing one translation over another. The better translations offer footnotes with alternate readings so that the reader is aware of the ambiguity.
Modern translations into English
No translation prior to the 1950s can be considered up-to-date with modern Biblical scholarship, for the simple reason that the Dead Sea Scrolls were not available to translators before then. Since the 1950s, there have been numerous translations made — too numerous to list here — with varying advantages and disadvantages.
Generally, translations are classified on a spectrum of formal equialence vs. dynamic equivalence. Strict formal equivalence would produce unintelligible English, because Hebrew and Greek grammar is so drastically different from English, so formally equivalent translations tend to take just enough liberties to produce grammatically coherent English sentences.
Dynamic equivalence tries to convey the same meaning in natural English, which necessarily involves different phrasing in some cases. For example, in Luke 15:8, the Greek word δρᾰχμή refers to a drachma, an ancient Greek coin, but such currency is totally unknown to most modern readers. Modern translations, even formal translations, generally replace it with "silver coin", although formal translations tend to explain the original word as a footnote. In other cases, ancient idioms may need to be rephrased to make intuitive sense to modern readers, where a formally equivalent translation may leave them as is.
Another difference between translations is how they handle the deuterocanonical books. It is not always practical to provide these in a way that is useful for both Catholics and Protestants. The book of Esther, in particular, has whole new sections added to it in the Septuagint, that — if included separately — would make reading Greek Esther quite cumbersome indeed. Instead, Catholic Bibles either replace translations of the Hebrew books with the Greek ones, or include the Hebrew Esther and the Greek Esther separately.
My opinion
I have been researching available translations and selected personal favourites from the available options. The New Revised Standard Version (NRSV) is generally accepted by Biblical scholars as the most accurate formal translation, and I now own a copy of the New Oxford Annotated Bible (NOAB), which is based on the NRSV and has a large number of annotations, maps and essays to explain how to interpret the scriptures. I love the NOAB as a reference, especially since it includes all deuterocanonical books, even those used by Orthodox churches but not the Catholic Church.
But the NRSV is not very amenable to easy reading, so I also now have a copy of the Catholic edition of the Good News Bible (GNB), which is written in clear, simple English, liberally seasoned with footnotes. I have been reading this for the past couple of weeks, and so far it is by far the smoothest translation I have ever laid eyes on.
Although it is quite popular, I do not like the New International Version, partly because of its Christian theological spin on the Old Testament (it uses "virgin" in Isaiah 7:14) and partly because it has fewer footnotes to explain alternate readings than the GNB. The KJV is right out for my purposes, as it is simply too old to benefit from modern Biblical scholarship.
But most of all, I'm glad I have researched the subject so that I have an understanding of how and why modern Bible translations differ, and what to expect. I have never actually read more than fragments of the Bible before, and the Good News Bible is making it extremely easy to do so, so I would wholeheartedly recommend that to any fellow Bible-curious folks.