Crochet is a method of creating fabric from yarn or thread, by using a hook and a ball of yarn. The hook is generally held in the right hand as shown. Even lefties can learn to crochet this way or they can reverse the instructions. There are two basic "holds".
In this method, you hold the hook as you would a pencil, grasping the hook between your thumb and index finger.
In the knife method, place your right hand over the hook and grasp it between your thumb and index finger to begin. The left hand is used to control the yarn and to hold the stitches as they are created.
Use whichever method feels more comfortable to you.