It might help to understand degrees of freedom in a more general context. In a physical system, the number of degrees of freedom is the number of things that have to be specified for me to know the complete state of the system--for me to know "everything there is to know" about it.

Imagine I've got a featureless point particle floating in a box. How many numbers do I need to write down to specify everything there is to know about the system? Well to begin with, I need to know where the particle is in the box, so I need three numbers: one for the x dimension, one for y, and one for z. That's three degrees of freedom.

Does that tell me everything there is to know? If the system is an unchanging snapshot sure, but not in a real dynamical system. If the particle can move, we need three more numbers: one for its velocity in x, velocity in y, and velocity in z. If you have those three numbers, you can predict where the particle will be at any given time, assuming you know where it started.

So, we'd say this particular system has six degrees of freedom: position in x, y, and z, and velocity in x, y, and z. There are six parameters that can vary to make a unique state of the system, given the constraints we've put on it.