The "degrees of freedom" are the independent pieces of information needed to exactly specify the state of a system. For instance, a particle moving in three dimensional space has 6 degrees of freedom.

Three numbers are needed to describe the position of the particle in space. (The x, y, and z coordinates do nicely.) Another three numbers are required to specify the velocity of the particle. (The components of the velocity vector in the x, y, and z directions for instance.) Thus, we need 6 numbers to exactly describe the particle's state.

There are many other situations where we can talk about "degrees of freedom". For instance, we may be studying the relationship between age, sex, and college GPA in a population. In this situation, we have three "degrees of freedom" for each person.