These four psalms, along with psalm 65, are called "songs." The headings for Psalms 68 and 69 specify that they are "of David."
Psalm 66 is something of a jumble. Verses 1-4 call the who earth to praise God. Verses 5-7 thank God for the parting of the Red Sea in Exodus. Verses 8-12 talk about God "testing" Israel, a reference, perhaps, to conquest and exile. Verses 13-15 promise God an animal sacrifice. In the remaining verses, 16-20, an individual declares to anyone who will listen that God has answered a prayer.
Psalm 67 has two strophes punctuated with the refrain,
May the peoples praise you, O God,
May all the peoples praise you.
This psalm declares that God has blessed Israel in order that Israel might make God known to all nations.
Psalm 68 is more of a mess than Psalm 66. I can't make much of it. Especially verses like this:
The women at home divide the spoil,
though they stay among the sheepfolds—
the wings of a dove covered with silver,
its pinions with green gold. (12b-13).
I know its a song of victory, but what does it mean? Verse 18 is quoted, to odd effect in Ephesians 4:8.
You ascended the high mount,
leading captives in your train
and receiving gifts from people,
even from those who rebel
against the Lord God's abiding there.
Granted, Ephesians quotes from the Septuagint, but still, its an odd use of an odd psalm.
Psalm 69 is an individual prayer for rescue from enemies who make false accusations. The notes in the New Interpreters Study Bible say,
In the NT, Ps 69 achieves kergymatic standing with Isa 53 and Pss 22 and 118.
In other words, New Testament writers applied this psalm's image of an innocent sufferer to Jesus.
Next: Psalms 70-73