Darning is a type of mending technique using needle and thread to cover a hole in a piece of fabric or reinforce a threadbare piece of fabric. Another type of mending is patching - using fabric in conjunction with needle and thread to repair holes and/or threadbare fabric.

How to darn a hole:

Clip away ragged edges
Work on the wrong side of the fabric.
Darn should be same shape as the hole but overlap the thinning fabric completely to reinforce
Use a darning egg (a light bulb or apple works too!)
Use a similar size yarn as is in the fabric you are darning
Work up and down the hole first (do not knot your darning thread)
Darn over the hole horizontally and only a couple stitches over onto the fabric (not as far out as your first darning stitches)
Pick up alternate darning yarns in each row in order to weave the horizontal darns through the vertical darns

How to darn a hole on knits:

Darning knits begins the same way as darning a hole in a woven fabric. The difference is that your second darn is diagonal instead of horizontal. This allows the fabric to stretch better once it's darned.

Here are some of my favorite resources:

Fab Scrap - a textile recycle resource (they pick up manufacturing scraps and sell them to artists and crafters) + great place to grab scraps for patching and mending
The Geometry of Hand-sewing by Natalie Chanin - a gorgeous and instructive book about hand-sewing
Complete Guide to Needlework - my favorite book for needlework since childhood
Make Do and Mend - a collection of 1940s British Government Leaflets centered around being thrifty in all sorts of ways including mending and repairing clothes and home goods
Pinterest mending idea board - a collection of images and videos related to mending