The number of degrees-of-freedom of a mechanical system is equal to the minimum number of independent co-ordinates required to define completely the positions of all parts of the system at any instant of time. This is a fundamental definition when thinking about the motion or vibration of a system.
In general, it is equal to the number of independent displacements that are possible.
A spring-mass which cannot rotate or move horizontally is said to have one degree of freedom (in the vertical direction).
An unconstrained mass has six degrees of freedom (3 translations and 3 rotations). A continuous system such as a beam has an infinite number of degrees of freedom. A system of particles could have up to 6 degrees of freedom, but would normally have less because of constraints (e.g. some of them might be joined by rigid links).
Statistics
Degrees of freedom is a term used in statistics to characterize the number of independent pieces of information contained in a statistic. For example, if we begin with a random sample of n observations and estimate the mean by the sample average, we are left with only (n-1) independent measurements from which to estimate the variance or deviations around the mean. In a simple regression, where we estimate both an intercept and a slope, only (n-2) degrees of freedom remain to measure variability around the fitted line.
Milling machine stiffness is a fundamental parameter to consider when wanting the best machined finish. For this we need to think about the degrees of freedom of the system. Where possible we need to lock-out some freedom (eg z-axis travel), stiffen the structure (eg reduce overhangs) and add damping.