If I remember correctly, it has to do with the kind of constraints the system has. If the constraints are functions of coordinates only, the system is holonomic. This usually means the system is "simple" to analyze (mass point on a circle).
If the constraints cannot be expressed via coordinates only, but are functions of velocity or even worse, the system is non-holonomic. Then we expect the system to behave in am more complex way. (ball rolling on plane surface without slipping).