There are many types of seams, some more common than others. While the plain seam is the most common, it is also over-used by amateurs. A plain seam is only suitable for delicate fabrics that will not fray. If you're using a delicate fabric that might fray, the classic stitch & trim seam might be your best choice. It can really be used on any fabric weight, but those that fray easily would benefit from a different finish.
Like most seams, the stitch & trim begins with a plain seam. This is simple enough. Place your fabric with right sides together and pin. Put your fabric in the sewing machine and sew a straight line ⅝" from the raw edge of the fabric. Don't forget to backstitch for ¼" at both the beginning and end of your stitch so the thread doesn't unravel. Press the seam open with a warm iron so it lies flat for you.
Once you've done this, it's time to create the stitch & trim seam. Sew a line of straight stitching ¼" from the raw edge of the seam allowance. Do this to each side separately and press flat again. Make sure you backstitch to keep your stitching in place. It does you no good if it unravels as soon as you've pulled it from the machine.
Once you've flattened your seam, it's time to trim. Cut away the excess fabric at the raw edge of the seam allowance using a pair of sharp scissors. Get close, but not so close that you clip the stitching. Once you've trimmed, press the seam open again so it lies flat.
Mastering this seam allows you to move on to more complex seams such as French, bound, and Hong Kong seams.