Abbreviated DoF
An object in the physical world has up to 6 cartesian DoF. [x, y, z] and [pitch, yaw, roll].
A mechanical DoF refers to the number of points of actuation a robot has. A robot arm with 3 joints and 2 extenders has 5 degrees of freedom, for instance.
Carteisan DoF are easily limited - the end effector must exist only within this plane (limiting to just 5 degrees of freedom). Mechanical DoF are not easily limited in this way, but there are many complexities and challenges from these DoF best described mathematically, later. [link, probably]