On an aesthetic level, lawn edging is a line of demarcation that creates visual interest in a landscape by separating one segment of your yard from another. But it may also be functional, as when you erect a barrier between a lawn and an adjacent flower bed (in which case we could just as easily term it "garden edging") to keep grass stolons from creeping into the latter.
There are a number of different types of lawn edging. Maybe the most basic distinction we can make is that between lawn edging that consists of a trench versus the various types that form a barrier. That is, some landscapers create an edge simply by removing sod in a nice even line to create the desired border, using a spade or a power edger. This might seem an easy way to install an edge, but the problem lies in maintaining it (you'll have to re-dig the trench).
That's one reason why most people prefer barrier-style lawn edging. But then the question becomes, What choices do you have for materials and how do you decide between them?
First of all, let's consider some "natural" materials. Stone is one of your possible choices. It's durable and looks great. But it can be expensive (unless you have some lying around in your landscape). It's also relatively time-consuming to install, since you're handling numerous individual units.
Wood is another material that many folks find attractive. But untreated wood will rot quickly; that's why landscape timbers are treated. You can install landscape timber edging around flower beds. But due to health concerns over pressure-treated lumber, I would not use it around a vegetable garden.
Then there are manufactured materials from which to choose. While metal lawn edging doesn't have too bad a name in the gardening community, many people despise plastic edging as being too cheap-looking.
If we get beyond the jargon and think outside the box, we will also recognize that there's some overlap between garden edging and raised beds. The typical raised bed is a shallow frame made of wood (the "edging" component) that is later filled with soil; then plants are installed in the soil. Thus the order of the steps in the project is simply reversed, as compared to the steps in installing garden edging (where the soil, and often the plants as well, are in place first; then the wood or other material is installed).
I provide instructions for creating a couple of atypical raised beds in the following tutorials:
See also: What Are "Edging Plants"?