If you're talking about the little round piece about 3" in diameter with a (probably) square stub on the front of it that mounts on the front of the flywheel, that's the starter clutch, and those bearings are supposed to "jam" when the starter rope is pulled. When you pull the starter rope, it attempts to turn this assembly clockwise as you are looking at it. The little ball bearings jam in the inside to spin the crankshaft and flywheel. As the engine (hopefully) starts, the little ball bearings are thrown outward by centrifugal force and release the inner part of this starter clutch. This clutch will also allow the flywheel and crankshaft to spin freely as the engine stops. There is a small hole in the end of the clutch which needs a couple of drops of oil two or three times in the entire life of the engine.
With the starter housing and rope assembly removed, if you can pull the rope and it will turn the inside of the starter housing freely and return by spring power when released, it's okay. If you can turn the starter clutch shaft counter-clockwise freely but it jams when turned clockwise, it's okay.
Remove the sparkplug and see if you can spin the flywheel freely by hand. If you can, and nothing makes terrible noises, the insides of the engine are likely okay. Check for compression by putting your thumb on the spark plug hole and cranking the engine. If it builds up pressure, the compression is okay, meaning the rings and valves are working. Next check for spark. If you've got compression and spark, it's likely a fuel problem.