Pivot pins are what permit a chairs mechanism to tilt back. It works like a hinge on a door. Usually, the pin has plastic bearings (non-replacable) to prevent noise and premature wear to the steel in which it is enclosed. If the bearings are worn out, metal will be worn away and the chair will rock and wobble, most noticeably from side to side. (Worn metal sometimes looks like grease on the floor or a powder).
When replacing a mechanism, mark the pivot location of the original mechanism on the seat bottom. Then place the pivot of the replacement mechanism near to the location mark.