The background pattern uses something called a 'plaited' or 'twisted' stitch. You wrap the yarn as if to purl, but knit it instead.
Row one: Knit one, then wrap yarn as if to purl and knit into the next stitch. Repeat across.
Row two: Purl in the stitches that you knit in the last row, knit in the others.
It should produce a subtle ribbed pattern. You may want to do a practice swatch (maybe 10 rows of 20 stitches each) before casting on the whole lot of 60 stitches
The knight and the rook are done in stockinette stitch, with the smooth side to the front. (M re and M li are knit and purl respectively.) The color changes are a technique called 'Intarsia' - video tutorial below. You'll want to use a second ball of your background color for the parts where the background splits around the figures, rather than carrying it across.
The Häkelnadel is a crochet hook, and it looks like it's only used for the fringe. I think the Stopfnadel would be a regular yarn needle for weaving in the ends. It's not a specific knitting needle.