A collet is a subtype of chuck that forms a collar around an object to be held and exerts a strong clamping force on the object when it is tightened. It may be used to hold a workpiece or a tool.
A chuck is a specialized type of clamp. It is used to hold an object with radial symmetry, especially a cylinder. In drills and mills it holds the rotating tool whereas in lathes it holds the rotating workpiece.
Pull Studs (also called retention knobs) keep the tool holder in the spindle. Using worn pull studs or using the wrong pull stud for your machine may cause the tool holder to suddenly fly out of the spindle.