Psalms

Most scholars recognize three primary form of poems:

laments,
thanksgiving and
hymn.

Each of these genre include specialized types. Penitential (seeking forgiveness) and Imprecatory (cursing) Psalms are specialized laments. Salvation History and Psalms of Trust are forms of thanksgiving.

Another way scholars group the psalms are by topic rather than form. The most common topics are:

Liturgical,
Royal,
Covenant,
Songs of Zion,
Wisdom,
Poems of the Law and
Messianic

Dennis Bratcher has made a table showing many of the genres of psalms.

There are various structural devices that can be used to better understand and categorize psalms.

The meaningful center is an important key structural device in many of the songs. The center word, phrase or verse is often a summary of the overall theme of the poem.

There are two common compositional devices the authors use to structure the songs:

Alphabetic Acrostic
Menorah Pattern - poems using 7 items in a 3-1-3 pattern in imitation of the Jewish menorah used in the temple.

Although not accepted by most biblical scholars, some scholars have also argued, quite persuasively, that many of the psalms are organized based upon a very carefully designed numerical structure. Casper Labuschagne's General Introduction to Logotechnical Analysis gives a rather detailed description of the use of numbers to structure the Old Testament content.

The editors of the Psalms divided it into five books, each of which closes with a doxology. There is no agreed upon reason for the number of divisions or why certain psalms are in each book. Commentators traditionally have drawn parallels between the 5 books of Moses and the 5 books of the psalms. Dr. Ernest Martin also argues for a parallel with a five-fold division of the Old Testament called the Megillot. You can read his complete description here.

Psalms	Law	Megillot	Season
Book 1	Genesis	Song of Songs	Passover
Book 2	Exodus	Ruth	Pentecost
Book 3	Leviticus	Lamentations	9th/10th of Ab
Book 4	Numbers	Ecclesiastes	Tabernacles
Book 5	Deuteronomy	Esther	Purim

Book 1 is made up of Psalms 1-41 and is comprised almost exclusively of songs written by David. The first two appear to be an introduction to the Davidic cycle of Book 1. As introductory poems they set up the theme of the ideal king, David, who delights in God's law and is specifically anointed by God as His unique representative king.

Throughout these psalms a few themes stand out:

Happiness is found in God's word and living it out through compassion to others. Happy or "blessed" is the first word in both Psalm 1 and Psalm 41, forming bookends to the collection.
David's firm trust in God. Over and over in these psalms we see David crying out for God's help and mercy, yet never doubting that God will eventually come to his rescue.

Below are links to posts on each of the psalms in Book I.