For the purpose of discussion, New Song is one of the many contemporary phrases that allude to what was called "singing in the spirit" for many years in charismatic circles of the 70s, 80s, and 90s. The scriptural bases for it are found in Ephesians 5 where we're told to all be coming together with songs, hymns, and spiritual songs. That last one is what some folks interpret as spontaneous song from the heart, unrehearsed, unplanned, but expected. Some commentators use the word "ecstatic" when talking about these songs. The phrase "new song" is found in several psalms (depending on the translation used) including 33:3, 40:3, 96:1, 98:1, 144:9, and 149:1. Isaiah 42:10 brings us the imperative: Sing to the LORD a new song, Sing His praise from the end of the earth! And finally, in Revelation (5:9, 14:3), "they sang a new song...", they being the saints triumphant.
What is it? Spontaneous, heartfelt expressions of adoration, praise, or worship is about as close as I can come quickly. Most of these are indeed "sung" as you and I understand singing, so far as I can tell. These can be nicely formed in verses with rhymes. More typically in my experience, they are more prose-like, with declarations and proclamations of Who He is, What He's like, my responses to Him. "I love you, Lord" is a good, common way to start it out. Picking scriptural phrases describing Him, proclaiming some aspect of Him that is particularly meaningful to you. That's all in your given language, of course. There is also the possibility of singing in tongues or in your prayer language, if you have been so gifted, and if that's a comfortable thing for your congregation. More than a few charismatics have been "let go" after introducing New Song to a baptistic congregation and forgetting to sing only in understandable syllables. Sad but true.
We were privileged to receive a nifty chart of chord progressions a bunch of years ago, with some nifty teaching on developing congregational participation in the New Song. It's good to start simply, maybe a two chord progression, with some instruction so that the folks know what they can do and that it's okay! Permission is important. Going until it's done is important, even when it seems to want to trickle off. Press in, press on! It's a journey, and it takes some practice.
For one's personal time, these can afford a nice foundation to explore New Song, both vocally and instrumentally. Remember the scriptures that talk about prophesying on instruments??? Numbers of folks think that's related to New Song. Some folks have taken the progressions, created different styles or beats with them using MIDI equipment to form "loops", and can practice without having a band, but still with the benefits of a larger sound. No hurting fingers from hours on the strings, and it can be done in any of the keys easily. Lets the mind and spirit soar without the limitations of musical abilities.