There is no biblical mandate that says we must only sing Psalms. Singing worship together goes back thousands of years to the Israelites who sang to the Lord (Judges 5) to the early church described in the New Testament (1 Timothy 3:16; 2 Timothy 2:11–13). Moreover, in Ephesians 5:18–21, Paul gives instruction for the church to address each other in psalms, hymns, and spiritual songs—emphasizing the variety by which we can express thanks and praise to God. Paul then restates these three forms of praise in Colossians 3:16. In James 5:13, James shares a similar sentiment, stating that if someone is cheerful they should sing praise to God. Ultimately, the Bible encourages creativity and variety in our worship of God. In fact, Psalm 96:1 encourages us to sing new songs to God!
Exclusive psalmody limits musical worship in a church service to only the Psalms. Adherents say the inspired words of God (the Bible) should always trump the words of people, no matter how pious, God-honoring, or "inspired" they may be. Churches which ascribe to this prohibit using any songs that do not originate with scriptural wording within the book of Psalms. Such churches include some Reformed groups such as the Free Church in Scotland and the Reformed Presbyterian Church of North America.
While singing the Psalms and other Scriptures is allowed in a church setting it is nowhere biblically mandated as the only mode of worship. In fact, Scripture indicates that different forms of worship are excellent and even encouraged, as seen in Ephesians 5:18–21 and Colossians 3:16. While exclusive psalmody claims the Psalms are only words Christians can use for their worship times, this is simply not biblical. The imposed restriction against Christians exercising their skills of composing worship songs and God-honoring lyrics is not biblically based, either.
The purpose of worship is to pray, teach the Word, sing, fellowship regularly, and such. Scripture doesn't mandate where or when this is to happen, nor does Scripture regulate such facets as the order of service, offering practices, and the origin of worship songs.