If you want to create a well-defined, discrete border around a curved flower bed, plastic edging is a simple material to use. It bends easily to match the curved contours, and because it's plastic, it won't degrade and will keep the borders looking sharp for years. Plastic edging comes in 20-foot sections that join together with couplers to form a continuous edge. You'll need stakes to support the edging, and while they may come with the edging, it's a good idea to have more than you think you'll need, especially if the garden bed is especially curvy.
Mark the boundary of the garden with spray paint. Make a continuous, flowing line, using a spare piece of edging to help you, if necessary. Avoid making sharp curves; plastic edging doesn't bend easily around corners.
Hold a spade shovel with the front of the blade facing away from you and, standing on the lawn, dig into the ground about 6 inches. It helps to mark that distance on the shovel with tape so you can consistently dig to the same depth. Dig a trench around the outline in this way, depositing the dirt behind it.
Place the edging in the trench with the rounded top at ground level. Pack dirt in under the bottom of the edging as needed to keep the top edge straight.
Cut the edging, if necessary, with tin snips and connect sections with a coupler. When you're joining sections, cut 1 inch of the rounded part of the edging off of each section and push the coupler into one of the sections, about halfway. Fit the rounded section of the other section onto the coupler and push the sections together. The flat parts will overlap.
Drive stakes at 3- to 4-foot intervals into the lower outside edge of the edging, using a 16-ounce hammer. The tops of the stakes should be slightly angled toward the lawn. Space the stakes closer together to support the edging around curves.
Backfill the trench with the dirt you removed from it, pushing it in behind the edging and tamping it down with your foot. The dirt should force the edging against the sides of the trench. Water the backfilled dirt with a garden hose to further compress it.