Hand darning is a technique used to fill holes in clothes or linens. Basically you are weaving more cloth where it has ripped or worn down. You only need a few things to hand darn - needle, thread (I use embroidery floss. You can also buy wool thread made specifically for darning) and scissors. It is also helpful to have an object under the hole like my Easter darning egg. When I darned my kid's pants, I used a flat Lego so I wouldn't catch the fabric on the other side! Beeswax can also help stiffen thread to make it easier to thread the needle and make the thread less likely to tangle.
Most of my embroidery floss is composed of six strands. For cotton sock darning, I usually split the floss and use three strands. The first step in darning is to stich around the hole to reinforce the edges. If you are darning socks, you will not want to put a knot in the thread, but rather tuck the loose end into other threads. It is not comfortable to step on knots. The next step is to create long stiches with a very short in a perpendicular stich to keep the lines parallel. These stitches become the warp threads.
Once these threads are in place, you make a small stitch to the place you can begin start weaving over and under making weft threads. When you finish one end, you put in a small perpendicular stitch and go back in the opposite direction. You can find more details on how to darn on the Collette blog.
Finish off the thread ends by going weaving the end into the work you have done. And then continue to wear your socks!
