Usually people darn holes in socks and use some fancy devices, but I'm not getting that complicated. I'm focusing mainly on the stitch pattern for non-stretch, non-curved fabrics. This is fairly simple and easy with a bit of practice.
You will make rows of vertical running stitches. The fabric I was working with is somewhat oddly textured (and of course my holes aren't perfectly round) so I think the drawings make it much clearer. You want to make the rows close together and extend beyond the hole a little bit. I wasn't sure if I needed to double my thread - I tried single and double and with the small holes I was mending, the single thread seemed to work better. Also, I am working on the wrong side of the fabric.
Now, you will repeat these stitched horizontally, weaving under and over the vertical rows. Tie a knot at the end and snip. My example is a little rough, but it looks fine from the outside.
I actually hand-stitched several little colorful darned holes on this garment (viewed from the back side). Come back tomorrow to see this full jacket!
See my other Re-Vamping Vintage series: