The Psalms have long been the hymnal of Christian worship. Jesus and his disciples sang the psalms of the Hebrew Bible and the practice continued with Paul and other early followers of Christ. In fact, insofar as we can tell, Christians of the first two centuries used the Psalm more than any other book of the Christian Old Testament. As the Church continued to grow and other Christian liturgical materials appeared (for example, the Odes of Solomon and hymns of Ephrem and Ambrose), the Psalms continued to form the basis for much Christian worship. By the fourth and fifth centuries, numerous commentaries on the theological and historical meanings of the Psalms had appeared, further cementing the Psalms as the foundational source for Christian worship of God in Trinity.
When contemporary Protestants and Catholics look back at how the early Christians used the Psalms, however, they quickly notice something is amiss: sometimes the Psalms seem to have been numbered differently. Not all of the Psalms are numbered differently, just most of them. And not all early Christians follow the same system. Further, some modern Bibles actually number the Psalms differently as well, lining up with how early Christians like Augustine, Jerome, and Ambrose. Why is this?
The reason that some Christians (early and contemporary) have Bibles with differently numbered Psalms is because there are actually two ancient versions of the Psalms, each of which numbers them differently. The Psalms were originally written in Hebrew and used by the People of Israel. A couple of centuries after the Babylonian Exile, however, fewer Jews living in Israel could read Hebrew, and even fewer Jews living outside of Palestine knew how to make sense of the language of their heritage. At least partially in response to this, a group of scholars translated the Hebrew Bible into the new lingua franca: Greek. This translation became known as the Septuagint, also called the LXX, because 70 (or 72) translators worked on the project. The LXX became the Bible for Jews living in the Greco-Roman Empire, was likely the Bible Jesus would have read, and most definitely served as the Bible for the writers of the New Testament (i.e., when the New Testament writers quote the Old Testament, they are quoting the Septuagint version, not the Hebrew).
Now, how does all of this impact the Psalms? Well, when the Book of Psalms was being translated from Hebrew the Greek, some of the Psalms were shuffled. For example, what were Psalms 9 and 10 in the Hebrew became Psalm 9 in the Greek. Both versions end up with the same number of Psalms (150), but (with a few exceptions) the numbering of the Psalms ended up being “off” by a single number. Thus, what many of us think of as Psalm 23 (“The Lord is my Shepherd…) was, for Christians following the LXX version of the Psalms, actually Psalm 22.
Subsequent versions of the Psalms have addressed these differences in a number of ways. Saint Jerome’s translation of the Bible into Latin (the Vulgate), for instance, follows the Septuagint numbering system. For many years, Roman Catholic translations of the Bible into English followed Jerome’s numbering, although more recent Catholic Bibles have reverted to the Hebrew designations. Protestant Bible translations have almost always followed the Hebrew numbering of the Psalms; thus, contemporary Protestant translations accord with the numbering of contemporary Catholic translations. The Orthodox Church continues to follow the Greek system, meaning that contemporary Orthodox translations of the Bible number most Psalms differently than contemporary Protestant and Catholic Bibles (this may be important to remember if you ever hold an ecumenical Bible study).
In summary: the reason that the numbering of the Psalms is different for different Bibles (ancient and modern) is because some Bibles follow the Hebrew numbering system and others follow the Greek pattern. This is just one more reason that understanding where the Bible has come from can help us better understand the varieties of Christianity in today’s world.
 Larry W. Hurtado. The Earliest Christian Artifacts: Manuscripts and Christian Origins. Grant Rapids: Eerdmans, 2006. 27, 32-3.