Amongst other things, a feeler gauge can be used for checking: tappet clearances, spark plug gaps, distributor points, bearing clearances and piston ring gaps.
Tappet clearances
Tappets must be a set width from the valve stem in an engine to prevent collisions.
Spark plugs gaps
Spark plugs should be set so a spark is able to transfer energy from an ignition system to a combustion chamber.
Distributor points
Distributor points need to be set so they can send high voltage from an ignition system to the spark plugs in a specific firing order.
Bearing clearances
Bearings should have a specified clearance in-between their casing to allow the crankshaft to rotate efficiently.
Piston ring gaps
Piston ring gaps should be set so the piston works efficiently and there is no excess oil consumption or increased levels of blow by gas.