degrees of freedom
How many independent numbers do you need to say exactly where everything is? A free point particle in three dimensions needs three. A rigid body needs six: three to place its center and three to fix its orientation. Degrees of freedom count the independent ways a system can move.
For N particles moving in three dimensions subject to k independent holonomic constraints, the number of degrees of freedom is 3N - k, and this equals the number of generalized coordinates you must supply. Each degree of freedom contributes one second-order Euler-Lagrange equation of motion (equivalently two first-order equations in phase space).
Beware a subtlety in statistical mechanics: the equipartition theorem assigns (1/2) k_B T of energy per quadratic degree of freedom, where 'degree of freedom' there counts quadratic terms in the energy, which is why a diatomic gas is more intricate than simple counting suggests. And a nonholonomic constraint restricts accessible velocities without reducing the dimension of configuration space, so kinematic and configuration counts can differ.
A rigid body free in space has 6 degrees of freedom; a molecule of N atoms has 3N in total, split into translation, rotation, and vibration.
Six for a rigid body: three translational, three rotational.
Count degrees of freedom, not coordinates: with a nonholonomic constraint you may still need more coordinates than the system has independent modes of motion.