A question like yours, though a good question, would take a book to fully answer. However, I know that many people have similar questions and they need as simple of an answer as possible. Therefore, I am going to give it a try. In order to keep the discussion narrowed down a bit, I will specifically deal only with the New Testament.
- First, the original copy (sometimes called an autograph) of each book was written by its human author and sent to the person, people, church, or churches addressed. For instance, Paul wrote 1 Corinthians as an individual letter to the church at Corinth and sent it to them.
- Second, these letters or books were recognized for their value to Christians as a whole and people began making copies of them for their own church or for themselves. These copies would be passed around and read over and over again. As older copies wore out, newer copies were continually being made.
Some key churches began to make collections of different letters for the use of their preachers and members. These collections might not have all of the New Testament books as we know them today and might have others books that are not included in the Bible today (like books in a church library), but they formed a basis for later decisions.
- The winnowing process took place over many decades as arguments for and against certain books as being scripture were made by the Bible teachers of the day. Although most of the New Testament books were not disputed, a few were. Also, some people supported books that we would reject today. Since about 90% of the text of the Bible was accepted by practically everyone and is still in our New Testament today, the questions were not as serious as some would make us believe. With time, the 27 books that we have today in our New Testament were accepted by almost everyone.
- During the time that a few of the books were being contested, the New Testament went out in many ways. Copies were made in the original Greek language. Translations were made into numerous languages. Portions of scripture were put into lessons (kind of like Sunday School lessons). Christians authors were quoting the Bible in writing after writing. These are important because they are the sources that will be used later for the establishment of our modern Bible.
Now, we need to jump in time to the early 16th century. Because of the renewed interest in the ancient writings at that time, there was a new interest in the text of the Bible. The Old Testament had retained a stable text known as the Massoretic Hebrew Text. However, the New Testament did not have any sort of authoritative text that brought the entire New Testament together in the original Greek language. Erasmus and men who followed him sought to establish the text of the Greek New Testament. The problem was that the Greek New Testament was found in hundreds of different portions of the New Testament. Some of these copies included almost all of the New Testament, but most of them had only a small portion; some containing only a number of verses. And, because of errors of copying and other problems, these manuscripts contained many different readings in different places in the Bible. Erasmus was the first scholar to study many of the different manuscripts and come up with an authoritative text for the entire Greek New Testament. During his lifetime, he improved on this text and others after him developed it even more. The Hebrew Old Testament and the Greek New Testament now needed to be translated into English.
- William Tyndale made the first modern attempt to translate the Bible from the Hebrew and Greek into English. He finished the New Testament (1526), the Pentateuch, and some other portions of the Old Testament, but never got the chance to complete the entire Bible before he died a martyr's death. Others followed him and the entire Bible soon found its way into the English language. Then, in 1611, in culmination of the work before it and as a testimony to the wonderful preserving hand of God, the Lord gave to mankind the King James Bible. Many other English versions have followed, but none have lived up to the standard set by the King James Bible.