My Owl and the Pussy Cat pattern uses Swiss Darning otherwise known as Duplicate stitch to sew the cats onto the hat. Here is a quick Picture Tutorial on how to Swiss Darn
Swiss Darning allows you to go over the stitches you have done to make them a different colour or picture. Stranded knitting is often used to do this when knitting in the round, but in some patterns, like The Owl and the Pussy Cat one of mine, the spaces between the different colour sections are too big and so swiss darning is very helpful for adding the colour on afterwards.
I like to work up and down the rows of knitting but you can work across too. Find what seems easiest to you. One darning stitches covers a stocking stitch, what you see is the legs of the V, the yarn also goes across behind the next stitch to hold it in place. You can see a few done already in the picture below. The yarn is now below the stitch about to be covered.
I hope this is clear, if my wording is confusing try just looking at the pictures, it was actually harder to explain than I thought. If anything is not clear or you need help please feel free to email me cameronsallyjane (at) gmail.com