Harmonic Balancer

Bad vibration? Look at the crank pulley(harmonic balance), does it seem to be wobbling? These are two metal parts with a rubber damper in the middle, check for oil leaks that might have weakened the rubber over time. It does run right?

Yes he states you can start and run the car, he has already removed the balancer and it says in the ad "new head gasket"

So if I understand you, the balancer is off the car and is being sold as is? If you buy it, you will either need to fix it there or tow it. You can't put the serpentine belt on a motor with no pulley! This means no battery charging or waterpump to cool the engine, or power steering.

I do believe that the balancer is not off the car yet, the man however removed one of the motor mounts, but he states that we can start the car and hear it run but can't drive it. My son said that we should buy the part put it on the car, tighten the motor mount just to see if it is ok, it is a 1997 Acura, 106,000 miles. Why would someone have the head gasket changed if there was nothing wrong with the car?

You wouldnt! It's not a preventive maintenance item. The head gasket must have blown for it to be replaced, a block test and or compression test is also wise b4 you buy it. Block test kits are sold at parts stores and detect hydrocarbon in the antifreeze, if they are there, the engine still has a problem, like a cracked head...