Hold the needle that has the cast-on stitches on it in your left hand. Hold the other needle in your right hand. With your yarn in front of the work, put the RHN from back to front into the first stitch on the LHN. Notice that the RHN is in front of the LHN.
Form a loop by wrapping the yarn on top of and around the RHN.
Bring the RHN under the LHN and carefully pull the loop through the stitch with the RHN onto the LHN to make a new purl stitch. Notice that the RHN will slip behind the LHN as shown.
With the new stitch securely on the RHN, slip the first or "old" purl stitch over and off the tip of the LHN. The purl stitch is complete. Notice that, unlike the smooth knit stitch, the purl stitch has a bump or ridge at its base.