The New Testament does not have a whole lot to say about music specifically, but the two primary passages that do, Ephesians 5:19 and Colossians 3:16, have certainly created a lot of debate and speculation. In particular, Christians have long puzzled over the meaning of the terms psalms, hymns, and spiritual songs in these passages.
In the most recent volume of Artistic Theologian, a peer-reviewed journal I edit, I wrote an article that assesses the current status of the debate over these terms.
Since ψαλμοῖς, ὕμνοις, and ᾠδαῖς are each used as translations of psalm titles in the LXX and are employed interchangeably in the NT, the weight of the evidence seems to suggest that Paul did not intend the terms to designate clearly identifiable genre of corporate song. Several implications can be drawn from this conclusion.
First, at very least these passages include a mandate to sing Spirit-inspired OT psalms. No matter how narrowly or broadly one interprets the terms, that Paul commands believers to sing psalms is clear. Whether these psalms are paraphrases or versifications is beyond Paul’s purview, but churches wishing to actively apply Paul’s instructions should make efforts to regularly incorporate OT psalms in their corporate repertory.
Second, conversely, no clear argument may be made from these passages alone concerning the warrant for singing songs beyond the OT psalms. Because these terms could refer only to different types of psalms, one cannot argue with certainty that Paul intended to broaden the church’s song beyond inspired psalms in these passages. Other NT passages may imply the allowance of non-inspired songs in Christian worship, but this cannot be proven from Ephesians 5:19 or Colossians 3:16.
Third, on the other hand, these passages do not clearly restrict Christian song to OT psalms. As with the previous point, the ambiguity of these terms presents enough uncertainty to prevent any dogmatic argument for or against a psalmody-only position.
Fourth, these passages are not relevant as defense for any side of the contemporary worship debates. Any attempt to define these terms using contemporary categories is anachronistic at best. No warrant exists to use these passages to defend contemporary praise choruses or the continuation of Spirit-inspired songs, but neither do these passages disallow them.
The only certain application to Christian churches from this phrase is that God expects his people to sing—at the very least they should sing inspired psalms.