Overlock

The overlock stitch is a popular, professional-looking way of finishing your raw edge. It's like a zig-zag stitch except each "zig" and "zag" are multiple stitches, and the exterior point will actually leave the raw edge to create a "lock" on the edge of the fabric. While most people think of Overlocking as something that is done on a serger or other high-end sewing equipment, in truth, if your sewing machine can do a buttonhole and a zipper, it can probably do an overlock.
Machine

In order to do the overlock stitch on your machine, you need a machine that has the overlock stitch setting, and you need the overlock presser foot. The distinguishing feature of the overlock foot is there will be a very tiny bar running down the middle of the needle's hole near the right side. You want to line this little bar up against the raw edge of the fabric as you feed it through the machine. The needle will actually jump over this bar on it's last stitch on the outside.

Not to scale, obviously

Different machines will have different stitches for performing the overlock, check your instruction manual and your presser feet options.. The stitches you're looking for will have a zig-zag quality. The idea is that the point of the zig-zag will extend over the raw edge and create a thread lock and prevent fraying.

Hand
To hand-overlock, come in from the back on one end of your raw edge. After that, you will always put your needle in at the front, but when your needle is still going through the fabric, pause and loop your extra thread behind the needle. When you pull the needle through, it will create a locking thread along the raw edge of the fabric.