Can you define each kind of psalms and each kind of prayer?
Wow, those are detailed questions and answers. I think that you need to remember that you will choose the psalms and prayer based on the occasion. There are this many so that you will be able to find the one you need based on when you need it.
A psalm is a type of sacred song or poem, and the most famous psalms can be found in the Bible. Though there are many subcategories, the five that you mention are the most popular. Psalms of praise are also known as hymns, and were written in order to celebrate God. Psalms of wisdom have a moral or teach a lesson. The royal psalms are about kingship and how kings can help their people be closer to God. Psalms of thanksgiving are written to thank God for something he has done. Finally, psalms of lament are about a painful experience or event. They are written as a way of bearing the pain associated with that event.
The types of prayer you refer to are those typically recognized by the Catholic church, though there are certainly many different types and ways to pray. In Catholic doctrine, prayers of adoration are similar to psalms of praise; they pay tribute to the wonder of God. Prayers of contrition allow a person to acknowledge their sins and ask for forgiveness. Prayers of thanksgiving, like psalms of thanksgiving, are a way of saying "thanks" to God. Prayers of supplication or petition are the most common; they allow a person to ask God for help with something.
The book of Psalms contains 150 poems meant to be heard out loud and maybe set to music as part of a religious service. Traditionally, they were thought to have been written by David, but modern scholars tend to think that they were composed by many different individuals over the course of many years.
Psalms of praise generally focus on the nature of God and not on specific things He has done. They tell about God's attributes and encourage the believer to praise Him for his goodness, righteousness, judgement, and wisdom. They often refer to playing music or singing and rejoicing.
Psalms of wisdom contain teachings and wise advice that are similar to the book of Proverbs, but in the form of a psalm. In studying the Psalms, we must remember that they were originally intended to be heard aloud, and perhaps set to music. These general principles are meant to instruct believers on how to live a Godly life.
Royal Psalms express the belief that God does His will and good works through the authority a king. Sometimes, royal Psalms celebrate the coronation or good works of an earthlt king. Other Psalms compare God to an earthly king and even point to the coming of God's son, Jesus.
Psalms of thanksgiving offer the believer's praise and thanksgiving for what God has done for him or her. Sometimes this is a specific good that God has granted, but sometimes thanksgiving for God's goodness in general.
Psalms of lament respond to a crisis, grief, or despair. They are intended to move a believer from hurt to joy. This type of Psalm expresses that God is present even when human beings cannot discern His presence in the world due to their own sufferings.
Adoration prayers express the believer's love of God and reverence for His goodness, wisdom, might, power, love, awe-inspiring nature. This is sometimes considered the purest and highest form of prayer.
Contrition prayers express sorrow for sins that the believer has committed. They are an important part of Catholic history, often used in the sacrament of penance.
Prayers of thanksgiving express the believer's gratitude to God for specific blessings He has given to the individual or humanity, or for His goodness, holiness, and strength.
Prayers of supplication express the believer's wish that God provide him or her with something he or she needs or desires. This is the most common form of prayer. Often a prayer of supplication requests health and healing for the sick.
Since the others have done such a wonderful job of explaining what the Psalms represent I will give you some examples of the types of prayers.
Matthew 6:9-15 "After this manner therefore pray ye,"- Teaches man how to pray. Prayer of Supplication.
Matthew 9:18 "My daughter is now dead; but come and lay thy hand upon her, and she shall live. Prayer of Supplication . Asks Christ to have child live again.
Matthew 21:6-9 " "out of the mouths of babes and sucklings thou hast perfected praise." Adoration prayers. The people have placed Christ on the donkey and laid down palm leaves to honor him.
Matthew 26:39 ""O, my Father, if it be possible,
let this cup pass from me; never less not as I will, but as thou wilst." Prayer of Supplication. Christ asks God for help.
Mark 14:22-26 This were the statements made by Christ at the last supper. The apostles sang a prayer song. It is most likely to be a prayer of Thanksgiving.
Luke 1:46-55 Mary's Song of Praise." My soul glorifies the Lord and my spirit rejoices in God ... "
I hope this helps.