Your professor is talking about representations of the position and orientation of a rigid body. Those two things combined have exactly six degrees of freedom in 3D space. You need three degrees of freedom just to specify the location of a dimensionless point. But most rigid bodies are more interesting than that. We also are interested to know which way the body is oriented. You need three more degrees of freedom to specify orientation.
Imagine an airplane. Which direction is it facing (what is it's magnetic compass showing?), North? South? East? 272.8°? The compass heading counts for one degree of freedom. But then, is it banked to the left? is it banked to the right? That's another. And, is it nose-up? nose-down? That's a third. Add to that the latitude, longitude, and the altitude; and now you've got six numbers to account for.
If a robot needs to know the position and orientation of some rigid, physical object in its environment, it's the same deal. You need at least six numbers† to represent it.
† In practice, the software may use as many as sixteen numbers if it represents position and orientation as transformation matrices in a projective space. Or, it may use as few as seven numbers if it represents position as a vector and orientation as a quaternion. While it is possible for software to use only six numbers (three position coordinates, plus three Euler angles), that often ends up being more complicated in the long run, because the software then must employ special-case strategies to avoid singularities in the math. (see also, gimbal lock).
