This is the first part of a three-part tutorial on dual quaternion algebra applied to robotics, whose goal is to help readers grasp the main ideas behind dual quaternions and their application to robot kinematic modeling and control. Part I presents the fundamentals of dual quaternion algebra, starting from the basic definitions of complex numbers and rotations in the plane and then extending the idea to rigid motions in the tridimensional case by means of quaternions and dual quaternions. In Part II, the dual quaternion algebra is applied to the kinematic modeling of different types of robots, such as mobile robots, robot manipulators, cooperative systems, mobile manipulators, and humanoids. Finally, Part III presents several control laws that are useful to control robots such as the ones modeled in Part II. Along with a relatively formal presentation, all three parts present several examples and exercises that may help readers in the comprehension of this increasingly important topic.