One traditional division of the book of Psalms—often called “the psalter”—divides it into five sections, on analogy with the five books of Moses: Psalms 1-41, Psalms 42-72, Psalms 73-89, Psalms 90-106; and Psalms 107-150, with Psalm 150 being the closing doxology for the whole collection. Those who accept this division understand the first and second psalms to be an introduction to the psalter as a whole, so some manuscripts give the number “1″ to the psalm we number “3.” If you are reading a psalter and it the chapters and verses don’t line up with your expectations, see if adding 3 to the psalm number corrects things.
It is obvious that Psalms was created from previous collections of hymns. See the psalms of Asap (Psalms 73-83) and of Korah (Psalms 42, 44-49), as well as the “Songs of Ascents” (Psalms 120-124). The duplication of some psalms is further evidence that the collection we have was created from earlier collections. Compare Psalm 14 and Psalm 53, Psalm 70 and Psalm 40:13-17 and Psalm 108 and Psalms 57:7-11 and 60:5-12.
Traditionally, most of the psalms in book 1 (Psalms 1-41) are ascribed to David, and most of them address Yahweh. In contrast, in book 2 (Psalms 42-72), most of the psalms are addressed to Elohim and, though 18 of them are ascribed to David, the rest are ascribed to other authors. (Notice that Psalm 72:20 tells us that we have come to the end of the psalms of David, even though 18 psalms after that are attributed to him.) Book 3 (Psalms 73-89) includes only one psalm of David, and most of its contents are addressed to Elohim. Books 4 (Psalms 90-106) and 5 (Psalms 107-150) are a mixture of things, so it is difficult to pick out defining features. Some manuscripts include a 151st psalm, a hymn of David describing his fight with Goliath.