When I was growing up my mother never threw out a pair of socks just because we had worn a hole in one.  I can recall the basket that sat beside her rocking chair in which she kept the socks waiting to be mended, along with the yarns, threads, needles and darning egg she used o perform her magic!
I interviewed her recently to get her method down, as I might just decide to darn the hole in one of my socks one day!

Mom's Method

1. Find thread or yarn matching the sock in color and texture;
2. Choose a darning needle with the eye sufficiently big to accommodate the yarn or thread selected
3. Place the darning egg in the sock and position the hole over the egg. Please note, Mom’s darning egg is wood and has been around for a very long time.  If you do not have a darning egg, you can use a light bulb, but careful as they can break easily.
4. Thread the darning needle with the yarn or thread – do not knot the end. Knots are unnecessary as the weaving action will prevent unraveling, and knots will be uncomfortable when wearing the sock..
5. Start on either side of the hole. Take several small vertical running stitches in the intact fabric of the sock, about 1/2 inch to the left or right of the hole. You are creating a starting line from which your weave will begin. Turn the sock upside down and make another row of stitches next to the first.
6. Increase the number of running stitches you make as you come closer to the hole. When you reach the point at which the hole begins, your stitching line should extend from 1/2 inch above the hole to 1/2 inch below it. You are creating the vertical lines of your weave first.
7. Continue making vertical running stitches. When you reach the hole, take your thread or yarn over the hole and into the fabric on the other side, forming what resembles a vertical bridge over the hole. Stitching should extend 1/2 inch beyond the hole at both the top and bottom edges. Continue this until the entire hole has been covered with your vertical lines of stitching.
8. Your vertical lines should extended 1/2 inch beyond the hole on either side.
9. Cut thread.
10.  Now you will begin the weaving process. Thread your darning needle and begin your work 1/2 inch from either side of the hole at either the top or bottom. Take the threaded needle and weave it under and over the vertical threads that cover the hole (as well as the vertical threads that lie within 1/2 inch of the hole).
11. Working backward and forward, weave each horizontal line over and under the vertical lines of stitching. Turn the sock upside down once you reach the opposite end of the hole, and weave another yarn strand next to the first.
12. Remember to alternate with each horizontal line.  The over stitch should always be adjacent to the under stitch in the previous horizontal line.
13. Continue stitching back and forth until you've completely filled the hole.
14. Trim excess thread.
15. You have just darned your first sock!  Remove egg (or light bulb) from sock, and wash.