"What did Paul mean by psalms, hymns, and spiritual songs? No one is completely sure, but most scholars agree he seems to be encouraging diversity in the songs we use to praise God. Psalms might be referring to the book of songs called the Psalter, hymns might be referring to songs that praise Christ and provide instruction and teaching, and "spiritual songs" might refer to more spontaneous expressions. If that’s the case then Paul is encouraging us to sing ALL our songs - short, long, fast, slow, old, new - with gratefulness to God." - Bob Kauflin from Sovereign Grace Music in his book Worship Matters
Early American Presbyterians mostly sang just the psalms, it seems to me that today the opposite is true. Evangelical Christians focus on what we might call hymns and spiritual songs to the neglect of singing the Psalms themselves. We sing bits and pieces of psalms in some of our hymns and spiritual songs but we don't sing the psalms in their entirety. And yet Paul tells us to sing Psalms. Now, why does Paul tell us we should sing the Psalms? Well, an easy answer that comes to mind is that the Psalms are God’s word. We want to make sure we are singing things that are true to the Bible so that, as Paul says in verse 16, “the Word of Christ might dwell richly in us.” And you can’t go wrong singing the Psalms can you? Another reason we might sing the Psalms is because our Presbyterian mothers and fathers did. And so we are connecting ourselves with them. But there is a greater reason to sing the Psalms. Those mothers and fathers were pointing back to a much more profound reason - Jesus sang the Psalms.
12 Put on then, as God's chosen ones, holy and beloved, compassionate hearts, kindness, humility, meekness, and patience, 13 bearing with one another and, if one has a complaint against another, forgiving each other; as the Lord has forgiven you, so you also must forgive. 14 And above all these put on love, which binds everything together in perfect harmony. 15 And let the peace of Christ rule in your hearts, to which indeed you were called in one body. And be thankful. 16 Let the word of Christ dwell in you richly, teaching and admonishing one another in all wisdom, singing psalms and hymns and spiritual songs, with thankfulness in your hearts to God. 17 And whatever you do, in word or deed, do everything in the name of the Lord Jesus, giving thanks to God the Father through him. (ESV)