David is the central voice
David is closely linked with the Psalms because Scripture directly presents him as a singer, poet, and Spirit-led speaker. His final recorded words describe him as Israel’s beloved psalmist, someone through whom God gave words to his people (2 Samuel 23:1-2).
The New Testament also identifies David as the speaker behind specific psalms. Peter explains that David spoke prophetically about the Messiah in Psalm 16, and Jesus says David spoke by the Holy Spirit in Psalm 110 (Acts 2:25-31; Matthew 22:43-45). So David certainly wrote at least some psalms, and probably many.
Now these be the last words of David. David the son of Jesse said, and the man who was raised up on high, the anointed of the God of Jacob, and the sweet psalmist of Israel, said,
Psalms was not written by David alone
The traditional headings printed above many psalms identify other contributors as well. They associate Psalm 90 with Moses, some songs with Solomon, and others with Asaph, Ethan, and the sons of Korah. These headings are not part of the numbered verses, which is why they do not appear in every Bible citation system.
That wider picture fits Israel’s worshipping life. David appointed musicians for public worship, including Asaph and other Levites, and Scripture says they ministered with songs before the ark (1 Chronicles 16:4-7). Later, King Hezekiah told the Levites to praise the Lord using the words of David and Asaph (2 Chronicles 29:30).
Moreover Hezekiah the king and the princes commanded the Levites to sing praise unto the Lord with the words of David, and of Asaph the seer. And they sang praises with gladness, and they bowed their heads and worshipped.
Some writers remain unknown
Not every psalm gives a name. Several begin without any author identified in their traditional heading. That isn’t a problem to solve. It simply means that God has preserved prayers whose human writers are no longer known to us.
Anonymous psalms still speak clearly. Psalm 1 opens the book by contrasting the way of those who delight in God’s instruction with the way of those who reject it (Psalm 1:1-6). The point is not to admire the writer, but to hear God’s wisdom and respond.
A collection for every season
Psalms is a library of songs, laments, prayers, wisdom poems, and royal celebrations, not a single work written in one moment. Its five-part shape ends repeatedly in praise, and the final psalm calls everything with breath to praise the Lord (Psalm 41:13; Psalm 72:18-19; Psalm 89:52; Psalm 106:48; Psalm 150:6).
So the simple answer is that David wrote many psalms, but he was not the only author. Different voices gave God’s people words for fear, repentance, gratitude, confusion, and joy. As you read a psalm, notice its setting if one is provided, but let the prayer itself lead you towards God.
Let every thing that hath breath praise the Lord. Praise ye the Lord.