The number of the canonical psalms is one hundred and fifty: but in the Septuagint version, as well as in the Syriac, Arabic, and Ethiopic translations, there is extant another, which is numbered CLI. Its subject is the combat of David with Goliath (related in 1 Sam. xvii), but it is evidently spurious; for, besides that it possesses not a particle of David's genius and style, it never was extant in the Hebrew, and has been uniformly rejected by the fathers, and by every council that has been held in the Christian church.
Sabine Baring-Gould, in "Legends of Old Testament Characters," provides the following translation and commentary:
PSALM CLI. (Pusillus erani).
1. I was small among my brethren; and growing up in my father's house, I kept my father's sheep.
2. My hands made the organ: and my ringers shaped the Psaltery.
3. And who declared unto my Lord! He, the Lord, He heard all things.
4. He sent His angel, and He took me from my father's sheep; He anointed me in mercy with His unction.
5. Great and goodly are my brethren: but with them the Lord was not well pleased.
6. I went to meet the stranger: and he cursed me by all his idols.
7. But I smote off his head with his own drawn sword: and I blotted out the reproach of Israel.
This simple and beautiful psalm does not exist in Hebrew, but is found in Greek, in some psalters of the Septuagint version, headed "A Psalm of David when he had slain Goliath." S. Athanasius mentions it with praise, in his address to Marcellinus on the Interpretation of the Psalms, and in the Synopsis of Holy Scripture. It was versified in Greek in A.D. 360, by Apollinarius Alexandrinus.
Andrew Adward Breen in his "General and Critical Introduction to the Study of Holy Scripture" writes:
In editions of the Greek text of the Old Testament, we find the CLI. Psalm attributed to David. St. Athanasius (Epist. ad Marcell. 15) and Euthemius (In Ps. Proem.) regarded it as authentic. The import of the Psalm is to celebrate David's victory over Goliath. It was never received in the Latin version, but it has place in the Ethiopian, Armenian, Syriac, and Arabic. It is not lacking in grace of thought and diction, but not good authority warrants its inspiration.
(Breen also notes the existence of eighteen psalms purporting to be of Solomon, but which from their tone are evidently from at the period of the captivity at the earliest.)
Psalm 151 ends up being one of those great examples of textual criticism properly applied in the West, even while many other errors in the Septuagint (particularly in the Psalms) were overlooked in the West, especially in the popular Latin version (not Jerome's translation).
It's worth noting that Psalm 151 shows up in Codex Sinaiticus as though canonical, whereas in Alexandrinus, it shows up in an appendix with the ascription to David noted and its being "outside the number" is noted.
What may be of interest to Tridentine Roman Catholics is that Psalm 151 is found in the Old Latin version, it is not included in the Vulgate and not in the Nova Vulgata (post Vatican II Vulgate). Trent, you may recall, dogmatically adopted the Old Latin Version in all its books and all their parts (which would consequently seem to include Psalm 151, if that was a part of the Old Latin Version). Alternatively, one could simply view the decree as a denial of the correctness of the Protestant canon, which excluded the Deutero-canonical books and apocryphal portions of the canonical books from the canon.
There is an interesting historical footnote. Among the Dead Sea Scrolls was found a "slightly different version of this psalm, in Hebrew" according to William Lee Holladay's "The Psalms Through Three Thousand Years." The bottom line, however, is that the Psalm was evidently not considered canonical by the Jews who preserved the text of the Old Testament in its original language, and was even recognized as non-canonical by some of the ancient Greek-speaking Christians (as evidenced by Alexandrinus and other ancient examples of the Septuagint).