For thousands of years, God's people have almost always recognized that the Book of Psalms (a.k.a. "the Psalter") is a very special part of God's word. We know this from all the sermons and writings about the Psalms throughout church history, but we also see evidence of this in the fact that Jesus and the Apostles quote the Psalms more than any other book in the OT. One unique aspect about the psalms is that they are simultaneously God's inspired word to us, and they are also divinely inspired prayers for us to pray back to God. This is one reason that reading, singing, praying, and meditating on the psalms has always been a foundational part of worshiping and serving God.
We know from the gospels that Jesus prayed the Psalms all the time. In fact, Jesus was praying the Psalms while on the cross (Ps. 22:1; Ps. 31:5). Even though the psalms were written long before the coming of Jesus, the psalms are, first and foremost, the words of Jesus. Jesus is the Word of God, and so all of Scripture reveals Jesus and comes from Jesus. Heb. 2:11-12 (quoting Ps. 22:22) also indicates that Jesus is the original source of the Psalter and the one who leads God's people in singing the psalms: "For he who sanctifies and those who are sanctified all have one source. That is why he is not ashamed to call them brothers, 12 saying, 'I will tell of your name to my brothers; in the midst of the congregation I will sing your praise.'" The Hebrew word for psalm means a "a song with musical accompaniment", and Jesus is the one who even still leads the singing.
In addition, Christians recognize that all the psalms are all speaking about Jesus and are ultimately fulfilled in Him. In Luke 24:44, Jesus explained to his disciples that all of Scripture (including the psalms) are fulfilled in him: "Then he said to them, 'These are my words that I spoke to you while I was still with you, that everything written about me in the Law of Moses and the Prophets and the Psalms must be fulfilled.'" The psalms are the words of Jesus to us, the prayers of Jesus for us, and the prayers that we are called to pray in union with Jesus.
As you study any given psalm, it's helpful to consider the following questions:
1) How does this Psalm describe the life and work of Jesus? How is this Psalm fulfilled in Jesus?
2) Because I am united to Jesus by baptism, how is this Psalm to be fulfilled in my life or in the life of God's people around the world?NOTE: Even if a particular psalm has no apparent application to my personal life, we can still pray that psalm for our brothers and sisters in Christ to whom that psalm may be very applicable.
3) How does this Psalm teach me to pray in union with Jesus?