When you are sewing a garment, the process of facing is when you create a clean finished edge on the inside of the garment. This can be done by using bias tape, binding, or facings made from the same fabric as the garment. Facing is often used on necklines, armholes, and hemlines.
One reason you might choose to use facing rather than simply turning under the raw edge of your fabric and stitching it in place is that facing will give you a much cleaner finish. When you turn under and stitch a raw edge, that edge is likely to show through to the right side of your garment if it's not sewn perfectly flat. By contrast, facing lies flat against your body and provides a smooth transition between the wrong side of your fabric and the lining (if your garment has one).
Another advantage of using facing is that it can add structure and support to certain areas of your garment. This is especially true if you're using interfacing in conjunction with your facing. Interfacing is a type of fabric that's used to stiffen other fabrics; it's often used in collars and cuffs to make them stand up nicely. Adding interfacing to a facing will help keep it in place and looking sharp.
If you're interested in trying out some garments with facings, there are plenty of patterns available online and in sewing books. Start with something simple like a tank top or t-shirt; once you've mastered that, you can move on to more complicated garments like dresses and jackets.