First of all, I recommend squaring up your quilt before you begin. I may do a separate tutorial on this at some point, but basically just cut your quilt down to the desired size you want before you start attaching your binding, making sure that the edges are straight and the corners are square. This helps to make sure the sides of your finished quilt are nice and straight and at 90 degree angles to each other. Once your quilt is squared up, you're ready to start!
Step 1: Pin the starting edge of your binding (the side with the bias end) down to the side of the quilt, with the raw edge matched up with the edge of your quilt. You want to start in from one of the corners a bit by at least a few inches to avoid bulk at the corners. A few inches of your binding should be open at this point, you are only going to pin down the bottom layer of binding to the quilt. You'll only need a few pins to get started.
Step 3: After you've sewn about 5", backstitch and close the binding. Start stitching in the same place you left off, but this time with the binding closed. Backstitch at the beginning again. At this point, it helps to make sure you have your machine set to needle down, so that the needle stops in the down position when you stop stitching.
Step 4: Continue sewing with the same seam allowance (3/8" or 1/4") until you get a few inches from the corner.
Step 5: Measure in from the bottom edge of the quilt, and mark your seam allowance. After awhile you'll get familiar with how far in you need to be, and you won't have to mark it anymore if you don't want to.
Step 6: Continue sewing, stopping at the seam allowance mark you just made. Backstitch at the end and trim the threads.
Step 9: Begin sewing again at the point where you marked your seam allowance, backstitching at the beginning.
Step 11: Now we're going to get ready to close the binding. Place the loose end of the binding on top of the starting end of the binding. Take note of where the bottom bias edge is, as well as where your stitching begins through both layers of the binding. Cut the loose end of the binding about halfway between these two points.
That's it! Your binding is attached to your quilt! There will be a little pocket that you can stick your finger into, but it won't create any problems (fraying, etc) since the unfinished ends are so far inside the binding. I find this way much quicker than other methods of joining the two ends of the binding.
Other posts in this series:
Quilting Basics: Making Straight-Grain Binding