Good Mill Workholding ensures not only that your part is fixtured securely within the machine, but that is does not vibrate or “chatter” while being cut. The most common source of chatter is holding a part across a large gap, like a bridge. Additionally, rough or sawn-cut faces may not be adequate fixturing, so serrated or “snap” jaws are used to hold these irregular surfaces securely.
Lathe workholding is only dependent on 2 factors: diameter and the Length/Diameter Ratio, or L/D. The first is obvious but the second is a consequence of the part deflecting as it is being cut but the tool. Skinnier parts with higher L/D ratios deflect more, resulting in chatter. To compensate, a tailstock is used to support the other end while cutting.