The Latin language was the language of a group of people in the part of central Italy called Latium. It was their ordinary, everyday language, which they used to write letters, buy groceries, flirt with each other, and everything else you use language for. Latin arose in the period starting around 700 or 600 BC, and by 100 BC or so had reached essentially the form, called Classical Latin, that it retains today. You can read more in the Wikipedia article on the
History of Latin.
The Bible was actually written in Hebrew and Aramaic (the Old Testament) and Greek (the New Testament). None of the Bible was written in Latin, there is no evidence that Jesus ever spoke Latin, and none of Jesus’ sayings are preserved in Latin. It is theoretically possible that he spoke to Pontius Pilate (a Roman official) in Latin, but other than that it is generally agreed that his everyday language would have been Aramaic and he would have spoken Hebrew for religious use in the Temple. It is certainly true that all the New Testament records we have of what he said are in Greek. The Latin Vulgate is simply a translation of the original texts of the Bible into Latin, much like your modern English Bible is a translation of those same texts into English.