Let's start with the inner essence of worship and then work out to the more public expressions of worship services or daily acts of love, which Paul calls our "spiritual worship" (Romans 12:1).
The reason I make the distinction between the inner essence of worship and the external expression of it is because I think Jesus did in Matthew 15:8-9: "This people honors me with their lips, but their heart is far from me; in vain do they worship me."
For Jesus, this worship amounts to zero. That is what "vain" means. "In vain do they worship me." Zero. It is not worship. This is a zero worship. It is zero if there is no heart dimension to it. So, you can do as many deeds as you want and go to as many church services as you want and never be worshiping if it is all external and nothing is happening in your heart toward God. All true worship is in essence a matter of the heart. It is more, but it is not less.
Then the question becomes: What is this inner, authentic, godward experience of the heart that we call the essence of worship? Jesus pointed us toward an answer in John 4:23-24 when he said, "The hour is coming, and is now here, when the true worshipers will worship the Father in spirit and truth, for the Father is seeking such people to worship him. God is spirit, and those who worship him must worship in spirit and truth." Notice that worshiping in spirit is not contrasted with worshiping in the body or with the body. Instead, it is put alongside worshiping in truth.
So what would that mean, that we are to be spirit-worshipers, worshiping from the spirit, and truth, driven by truth? I think the point is that when we worship - right worship, good worship, pleasing worship - depends on a right mental grasp of the way God really is, truth.
If we worship an idol of our own creation, we are not really worshiping God.
Secondly, worship depends on a right spiritual or emotional or affectional heart-grasp of God's supreme value. So true worship is based on a right understanding of God's nature, and it is a right valuing of God's worth.
Of course, his worth is infinite. Thus, true worship is a valuing or a treasuring of God above all things. That would be the closest I am going to give to a definition, I suppose. True worship is a valuing or a treasuring of God above all things.
The inner essence of worship is the response of the heart to the knowledge of the mind when the mind is rightly understanding God and the heart is rightly valuing God.
Or you could use words like treasuring or prizing God or delighting in God or reverencing God or being satisfied with God. All of these inner responses to God reflect his infinite worth and beauty. That is what worship was designed to do: put the supreme worth of God on display.
In fact, the English word "worship" comes from worth ship. That is, worship is showing, displaying the worth of God.
Two Acts of Worship
We worship God authentically when we know him truly and treasure him duly. Then the word "worship" refers to that valuing, that inner valuing, becoming visible in the world in two basic ways in the New Testament.
One is acts of the mouth: acts of praise and repentance in worship services or small group gatherings.
The other is acts of love with the body and the hands and the feet: acts of love that show the supreme value of God by what we are willing to sacrifice for the good of others.
I get those two things from Hebrews 13:15-16. Listen to this amazing summary. It says, "Through him, then," - through Christ - "let us continually offer up a sacrifice of praise to God, that is, the fruit of lips that acknowledge his name. Do not neglect to do good and to share what you have, for such sacrifices are pleasing to God."
Those two verses begin and end with the term "sacrifice." Of course, the sacrifice is an echo from the Old Testament sacrifices which were at the center of the worship and were to display the value of God as we gave up a bull or a goat and showed that God is precious to us.
We value his redemption that comes to us through the sacrifice and now through the sacrifice of Christ. So through Christ two things become worshipful sacrifices in our life: the fruit of lips that acknowledge his name; that is, worship services in singing and praying and repenting and confessing, and secondly, the fruit of deeds.
Don't neglect to do good. Share what you have. Such sacrifices are pleasing to God. Both of those are acts of worship.
Heart of Worship
You see it again in Romans 12:1: "I appeal to you, therefore, brothers, by the mercies of God, present your bodies as a living sacrifice, holy and acceptable, which is your spiritual service of worship." So all of our bodily life done in love for other and in reliance upon God display the worth of God above all things and make us worshipers in our daily life.
So, here is my summary: The inner essence of worship is to know God truly and then respond from the heart to that knowledge by valuing God, treasuring God, prizing God, enjoying God, being satisfied with God above all earthly things. And then that deep, restful, joyful satisfaction in God overflows in demonstrable acts of praise from the lips and demonstrable acts of love in serving others for the sake of Christ.
Interview with John Piper (Founder & Teacher, desiringGod.org)