A hole in a sock cannot just be sewed together with a needle and thread, as this would cause a lump. A proper darned sock leaves no lump, the sock is comfortable to wear.
Some people would say why would you want to darn a sock, when you can just throw away socks with holes, and buy new ones? Have you ever had a pair of socks that fit so well and felt so right, not to tight, stays up and very comfortable? This would be one of those times to darn. How about a "security" sock, the ones your child just loves, or just a challenge, to see if you can do it? No matter what the reasons, you save yourself money, by repairing.
Darning socks is like trying to solve a puzzle. One must decide where to start and from there, the weaving begins, in and out until the patch is complete. Choose the yarn, wool that matches your sock in thickness and color. Use a round hard object (a round hard ball) to place inside the sock, this holds the sock in place while mending, this maintains the hole in one position, while you work on it. The round hard object also helps guide the needle end to slip smoothly across the hole. Start by turning the sock INSIDE-OUT and work in good light. When darning a hole in a sock, the hole is slowly filled by weaving yarn in and out, from one side of the hole, to the other side of the hole, creating a network of yarn stitches which cross to fill the hole.
For this examples, different color of wool are used, so you can see the stitching.