A little background is in order here. While the individual psalms themselves may be very early in origin, the final organization of the Psalter into 150 psalms took place sometime in the late post-exilic period. In fact, the earliest evidence we have of the canonical form of the Psalter is the Septuagint (LXX), the Greek translation of the Old Testament. The Septuagint dates back to the second or third century B.C.
The Masoretic Psalter (MT), in its final form, cannot be dated until much later. Elements of the eschatological hopes found in the Greek Psalter were omitted from the Masoteric text, reflecting the impact of the destruction of Jerusalem in 70 A.D.
However, despite minor variations, it is clear that the MT and the LXX contain the same psalms, placed in the same order, though their numbering is slightly different. The LXX combines MT Psalms 9-10 and 114-15, while it separates MT Psalms 116 and 147 into two.
The Catholic Church has always accepted the LXX as canonical, including a handful of books that are not found in the MT. Protestants generally use the MT and refer to the “extra” books in the LXX as “apocryphal,” while Catholics convincingly point to the historical and theological basis for accepting the LXX in its entirety. For more on that issue, click here.
Since both forms of the OT contain all the same Psalms in the same order, with only a difference in the numbering, most modern Catholic Bibles, in a spirit of ecumenism and good will, have adopted the numbering of the MT even though otherwise they preserve the LXX. An exception is the Douay-Rheims Bible, which retains the LXX numbering.
An example of this at work: There is the Miserere, the Psalm that begins “Have mercy on me, God, in your kindness; in your mercy blot out my offense . . .” That Psalm is generally referred to as Psalm 51, but in the Douay-Rheims (LXX), it is Psalm 50. For that reason, you will sometimes see this Psalm cited as Psalm 51 (50).