Hand-eye calibration is used to relate what the camera (“eye”) sees to where the robot arm (“hand”) moves.
In a nutshell:
Eye-in-hand calibration is a process for determining the relative position and orientation of a robot-mounted camera with respect to the robot's end-effector. It is usually done by capturing a set of images of a static object of known geometry with the robot arm located in a set of different positions and orientations.
Eye-to-hand calibration is a process for determining the position and orientation of a statically mounted camera with respect to the robot's base frame. It is usually done by placing an object of known geometry in the robot's gripper and taking a series of images of it in a set of different positions and orientations.
For a detailed description of the topic, please read the following pages: