As I write this, questions arise. Please correct me in responses.
Think of a "degree of freedom" as a coordinate. A point particle has three (x,y,z), two point particles have 6, etc. A finite rigid sphere has six, three coordinates of the center of mass and two angular coordinates (polar, azimuthal), pertaining to rotation of the sphere.
Nature is thought to tell us how many degrees of freedom there are. The equipartition theorem was frequently used in the nineteenth century-- 1/2 kT of kinetic energy with each degree of freedom associated with motion. When experiment suggested "the count was off" great discoveries (quantum mechanics!) were made.
Think about a diatomic molecule. Here the count is three degrees of freedom for translation, two for rotation. The bond vibration is "frozen out," and the spin parallel to the bond axis is "inactive", essentially becase the moment of inertia is no huge.
When we count 3N - 5 degrees of freedom for an N-nucleus molecule, the 5 omitted are three translations of the COM and two rigid body rotations.
John E. Reissner
The University of North Carolina at Pembroke
reissner@gmail.com