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-replaceable) to prevent noise and premature wear to the steel in which it is enclosed. If the bearings are worn out, squeaking is common and metal will be worn away causing the chair to rock and wobble most noticeably from side to side.
(Worn away metal sometimes looks like grease or a powder on the floor).

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.