These days many people think of the word "liturgy" and (if they know the word at all) immediately think of a "high church" or very formally-organized worship service. You will sometimes hear people refer to a church's worship style as "liturgical" which means (to them) that an order of worship was followed closely, probably that there was a fair amount of responsive portions from the congregation, maybe that the minister wore a robe, etc.
I understand that this is what many people think of when they encounter the idea of liturgy or of something being "liturgical"-- and to a certain extent, there's nothing wrong with using the term this way. But, like so many words and concepts, our world isn't using the word correctly, and that misuse keeps us from having a real understanding of what we are setting out to discuss.
"Liturgy" is not a complicated term. It simply means "public service."
When we have a liturgy in the church, it is a public service of worship. In other words, whatever you do as a part of the worship service* you participate in, that is the liturgy. If your church uses lots of responsive readings, unison prayers, and other highly-participative elements, that's the liturgy. If your church's worship is essentially just songs and teaching, then that is the liturgy.
Which is to say, unless your church's "worship" is nothing more than a concert that you sit and passively listen to, maybe with a teaching time inserted in the middle, then your church has a liturgy! (And those churches who DO present the concert with teaching inserted as worship, I would argue, are not engaging in biblical corporate worship. That's just entertainment.)
Therefore, it's not accurate (or even possible) to speak of a "liturgical" vs. a "non-liturgical" worship style. One worship style might be "more liturgical" than another, but even there I would challenge this assumption-- it implies that a more heavily-responsive liturgy is more actively engaging than a less heavily-responsive one, which may not actually be the case (though it if is, perhaps the worshiper should seek out the more engaging style if he is earnestly seeking to worship God).
"Liturgy" isn't anything mystical, legalistic, or archaic. There are churches that employ more highly-structured liturgies that are all of those; likewise, there are churches that employ a much more "freestyle" liturgy that are all of those, as well. We should learn to use the term "liturgy" for what it means, and recognize the benefit that our various liturgies are to our congregation's worship.
I'll get into how liturgy benefits us in a future post.
*I realize that I've inadvertently stumbled into using another term that is misused or misunderstood today: the idea of a worship service. Many churches don't use this term now, preferring "worship event" or some such; but that is another rant altogether. For now, please accept my premise that I use the phrase "worship service" because it is the term the Bible uses to describe our public act of worship.