task space
Task space is the way we describe where the robot's hand actually is and which way it is pointing, out in the physical world — not in terms of joint angles, but in terms a person would naturally use: this position, this orientation, in this room. A point in task space is the hand's full pose: where it sits in space (its place) together with how it is tilted and turned (its facing). It is also called operational space, because it is the space in which the actual job gets defined.
This is the language of goals. You almost never want to tell a robot raw joint numbers; you want to say put the gripper here, level, facing the shelf, or move the welding tip along this seam at this speed. All of those instructions live in task space, because they are about the hand's place and aim in the world, completely independent of how many joints the arm has or how they happen to be arranged inside.
Because the motors only understand joint space, the robot must constantly translate between the two. Going from joint angles to the hand's pose in task space is forward kinematics; going the other way, from a desired pose in task space back to the joint angles that achieve it, is inverse kinematics. Task space is where we state what we want; joint space is where the machine makes it happen.
Saying lift the spoon to the bowl, kept upright is a task-space command — it names where and how the hand should be, out in the world. Your brain quietly figures out the matching shoulder, elbow, and wrist angles, which are the joint-space answer.
We give goals in task space (where and how the hand goes); the body solves for the joint angles.
A full pose in task space usually needs six numbers: three for position and three for orientation.