How you group the psalms into types depends on how general or detailed you want to be.
I like a simple, three genre classification: praise and thanksgiving, lament, and wisdom.
Praise psalms typically begin with commands to praise like “praise the Lord” or “sing to the Lord.”
Lament psalms typically begin with cries like “O Lord,” before expressing the lamentation.
Wisdom psalms offer observations on how to be happy, how to live godly lives, and so on.
I group praise and thanksgiving together, but they are not identical. I’ve sometimes heard that we thank God for what He does and praise Him for Who He is. That is an oversimplification that does not always hold up.
Claus Westermann, in Praise and Lament in the Psalms, gives a more thoughtful distinction. Praise is a free and spontaneous expression. Thanks is more like an obligation. Praise is always public and vocal. Thanks can be private and silent. Praise elevates the one praised. Thanks does not.
In What the Old Testament Writers Really Cared About, John C. Crutchfield gives an eight-genre classification: lament, praise, royal, thanksgiving, trust, wisdom, liturgy, and historical. He also calls some psalms “mixed” and some “unclear (as to genre).”
They are hymns, wisdom, thanksgiving, royal, and torah to name a few.
2. Thanksgiving Psalms
4. Enthronement Psalms
5. Royal Psalms
6. Zion Psalms
7. Wisdom Psalms
8. Trust Psalms
10. Torah Psalms
The five main Psalm genres are as follows:
Smaller or mixed
These genres have some subgenres, which can be found in the reference link provided.