Double knitting is when you knit two sides of a double thick project at once, using two different colors of yarn. Following a color chart for this is an excellent brain exercise, as your background color and the color of your motif keep switching back and forth. When you knit on one side, the cup is purple, when you work your way back, the cup is black. Besides that, every time you work one color, you also work the other, and there are stitches in both colors on your needle at the same time, even when you are in the midst of a solid block of color.
How does it work?
You cast on with both colors at once, making two loops (one of each color) for every stitch cast on. Then you knit each stitch separately, knitting one color then purling the second. The knits become the front of one side of your project, and the purls are the back of the other side. The directions say, "As with any double-knitting project, be sure you’re bringing both yarns to the front and back as you knit one side and purl the other." What this means is that when you knit, both yarns are behind your work, and when you purl both yarns are in front. This seems counter-intuitive at first, but as your work progresses, you can see that the yarns actually end up in the middle between the two layers of fabric that are developing.
Notice in the pictures that you actually have twice as many stitches on your needle as on the face of your project. The extra stitches are making a mirror image on the reverse side.