joint space
Joint space is the robot's description of itself from the inside — a list of numbers, one for each joint, that says exactly how much each joint is currently turned or slid. If an arm has six joints, a single point in joint space is just six numbers (say, six angles); change any one number and the arm visibly moves. Every pose the arm can hold corresponds to one such list, and the whole collection of possible lists is the space.
Think of it as the robot's set of internal dials. The robot's motors and sensors live entirely in this world: a controller tells joint three to turn to 40 degrees, and a sensor reports back that joint five is at 12 degrees. None of these numbers say where the hand is in the room — that takes a separate calculation. They only describe the shape the arm is folded into, dial by dial.
Joint space is one of the two great languages of robot motion. It is the natural one for the machine, because joints are what the motors actually drive. Its counterpart, task space, describes where the hand is and how it is pointed out in the world — the language humans usually think in. Converting between the two, in both directions, is one of the central jobs of kinematics.
Your own arm in joint space might read: shoulder lifted 30 degrees, elbow bent 90 degrees, wrist twisted 10 degrees. Those three numbers fully fix the shape of your arm — yet on their own they never tell you where, in the room, your fingertip has ended up.
A handful of joint angles pins down the arm's shape, but says nothing about where the hand sits in the room.
The number of independently driven joints (and thus the number of dials in joint space) is the arm's degrees of freedom.