To darn is to stitch up a small hole in a piece of clothing. Instead of throwing your worn-out socks away, you can just darn the holes in their toes.
When you darn your socks or sweaters, you use a needle and thread to close small holes in the woven fabric. There's even a specific stitch known as a "darning stitch," in which you first weave the thread with the grain of the fabric, and then fill in the other "woven" direction. The result is a sturdy patch made only of thread. Darn comes from the Middle French darner, "mend."