What you need:
- piece of clothing you want to patch up
- patch fabric
- double-sided fusible interfacing (I've used Steam-a-Seam 2
1/2" wide tape for my patches.)
- iron & ironing board or towel
- sewing machine or needle-and-thread
For the patch fabric, pick something of similar weight and color/pattern to the clothing you want to patch, if you want the patch to blend in. If you want it to stand out, pick something bright and colorful.
Generally, knit fabric and some woven natural fabrics tend to not fray too badly. Synthetic or otherwise slippery wovens (like the heavy woven tie-silk I'm using here) tend to fray pretty badly. Using the double-sided fusible web tape helps keep those fabrics from fraying.