There have been many books written about the source manuscripts used in modern Bible translations, and I cannot match their depth or completeness. If you want to build an extensive knowledge of Biblical sources, read those books. A great starter guide is The New Evidence that Demands a Verdict, by Josh McDowell. Nearly everything I write here will be a recap of what I read in that book.
That said, I will start with an overview of the translation process, which is how the Bible gets from ancient manuscripts to the english (or other language) translations we have available today in bookstores around the world. Then I will give an overview of the different source materials available for the OT, and then the sources of the NT, as well as a brief description of how these sources play into different translations. I will conclude with a description of some of the different major Bible translations and the philosophies underlying those translations, which can result in even more differences in the same verses from version to version, simply based on how the words are interpreted.
Translating a Bible is not as simple as modern translations of books, or speeches or whatever, from one modern language to another. The biggest reason for this is that we do not have the original source texts, which I will henceforth refer to as "the autographs". The only extant documents are copies of the originals (which I will henceforth refer to as "sources" even though they aren't the original sources). As such, translating the Bible has three main steps. First, you have to gather all of the existing manuscript fragments together. Second, you have to collate all of those documents into a single master document. This is because the fragments are often of the same passage, but there are sometimes mistakes in some of the fragments. You have to weed out the mistakes by cross-referencing the different fragments against each other. And I am deliberate in using the word "fragment", because of all the extant manuscripts, almost all of them are just brief passages, like (for example) John 6:5-20. It's usually just a literal scrap of text, but there are thousands of them and when put together, you can get a whole copy of the Bible. But this leaves some serious challenges in resolving conflicts between different scraps. Usually these resolutions can be made on the basis of a variety of factors, but sometimes disagreements persist between different versions, and in a small handful of cases these disagreements are theologically significant. Third, and lastly, once a collated document has been created, you have to translate that from the source language into English (or whatever). Each one of these steps has taken thousands of people a collective hundreds of thousands of hours of scholarship over the past few thousand years, so suffice to say every single letter in the Bible has been scrutinized by hundreds of people all around the world ever since it was first written.
The first step is largely archaeology and is mostly removed from the textual/linguistic analysis that goes into the translation effort. The second and third steps are both immensely complicated endeavors that I cannot hope to describe in detail, so I think I will stop here and say, "read Josh McDowell if you want to know more. Or just search online for more."
Some additional difficulties faced by translators:
1) Some of the extant copies are not written in the original language of the autographs
2) The source manuscripts contain numerous obscure words for which there are no proper modern translations
3) Cultural differences in writing style from back then until now result in some issues that have to be glossed over by modern translations, because they would be confusing or misleading to modern readers. One example (among many) is units of measurement. The standard Hebrew measure of distance is a "cubit", which is somewhere around 1.5 feet. But for the translations to reference "cubits" all over the place would confuse a lot of people who are understandably unfamiliar with it, while it would be deeply familiar to ancient readers.
As I briefly alluded above, there are a variety of sources for the OT that come from many different periods and (perhaps more surprisingly) many different languages. I will start with the most important source manuscript of the OT, the Masoretic texts.
The Masoretic texts are a group of Hebrew documents dating to roughly 800 CE. This is, surprisingly, the earliest complete Hebrew OT that exists today. What's fascinating about this is that the earliest complete NT manuscripts are actually much earlier, dating to between 300 and 400 CE. So the documentation of the OT is generally worse than the NT. However, some evidence exists that shows the general accuracy of the Masoretic texts, so all is not lost for that reason. Jewish scholars over the millennia have demonstrated an almost fanatic devotion to accuracy is their retention of the OT. While this has not guaranteed the non-existence of any errors, it has made the situation a lot better than it is for many other ancient documents.
One of the biggest reasons scholars believe the Masoretic texts are accurate is cross-comparison with the Dead Sea scrolls. The Dead Sea scrolls are a set of various documents (not all Biblical) dating to roughly 100 BCE. While the date is great, the biggest problem is that they do not include a full copy of the OT, so we can only cross reference a subset of passages. One of the biggest discoveries is a complete copy of the book of Isaiah (which for those who don't know, is quite a long document). Compared with the Masoretic Isaiah, it was nearly identical except for a few trivial spelling differences.
Another source for the OT is the Septuagint, which is a Greek translation of the OT dating to around 70 BCE. This is useful for its date and its completeness, but unfortunately it is a translated work so (obviously) it is not in Hebrew. This adds a layer of difficulty for English translations because it essentially requires you to "double translate" the book, from the original Hebrew to the Greek Septuagint, and then from that Greek to modern English. Double-translating almost always introduces a lot of inaccuracy into the final product, due to all of the factors that make even a single translation being compounded through two languages. If you ever experiment with Google Translate you can see this for yourself.
Similar to the Septuagint is the Aramaic Targums, which are a set of Aramaic translations of various passages of the Hebrew OT. The Targums have a somewhat different character from the Septuagint, in that they tend to be more didactic and less literal, but are essentially just another ancient OT translation. They date to roughly 0 CE, plus or minus a hundred years.
Lastly (that I will mention), is the Vulgate. The Vulgate is a complete Bible translation of both the OT and the NT into Latin, and it dates to roughly 400 CE. This translation is significant because the OT was translated into Latin from the original Hebrew, much like the Septuagint of 500 years earlier. Research has shown that the Hebrew from which it is derived is largely in correspondence with the Masoretic texts, again validating their accuracy.
That concludes what I will say about OT manuscripts. For the NT, the situation is a bit different. For one, the set of documents we have are much closer to the autographs in terms of age. The earliest complete NT is from around 300ish CE, compared to the origination of between 50 and 90 CE for when the books were all first written. There is even an extant fragment of John dating to around 130 CE, which is remarkably early and would have existed in the lifetime of some of the witnesses of the Acts of the Apostles. You can find the wikipedia article on that fragment here.
Putting aside a lot of nuance and detail for the moment, nearly all of the major extant copies of the NT are found to belong into two large families of documents. These are groups of documents that almost completely agree within themselves, and mostly (but not entirely) agree with each other. They are called the Majority text and the Minority text, alternately called the Byzantine type and the Alexandrian type. The names are indicative. The Majority text is the statistically larger group of documents, while the Minority text is from a comparatively smaller group of documents. The Majority text is largely derived from a group of documents that were originally housed in Constantinople; hence, Byzantine. The Minority text is derived from documents originally housed in Alexandria, Egypt. The Majority text is much more widespread because it was largely adopted in substance by the Catholic church and it forms the basis of the King James Version (KJV) of the Bible. However, the Minority text represents more geographically and lexicographically dispersed documents, which is much more valuable for the purposes of critical analysis. As such, the Minority text has held increasing weight in recent decades compared to the long centuries of Majority dominance. For example, the NIV frequently holds to Minority text positions where the two versions don't agree.
These disagreements are relatively infrequent though, so the vast substance of the NT is not called into question by any differences between the two bodies of manuscripts.