Crankshafts vibrate with a twisting motion due the torque variation that results as each cylinder fires. This vibration is at its worst at some critical RPM. On a V8, this RPM is typically in the 4-5000 RPM range. If you operate the engine at this critical RPM without a harmonic balancer excessive vibration will eventually BREAK THE CRANKSHAFT. DON'T LEAVE IT OFF.
A harmonic balancer modifies the vibration characteristics of the crankshaft and flywheel assembly. They are made from of a heavy steel ring mounted over a thin layer of rubber on the outside of a center hub. The rubber acts as a spring. The ring twists on the rubber to cancel out crankshaft torsional vibration. Some energy is removed as heat in the rubber but the balancer's primary function is to store energy and return it to the crankshaft at just the right time to suppress vibration.
For you more technical types, this is a classic "dynamic vibration absorber". These are sometimes used other places in the car to solve vibration problems. Typically, if you find large chunks of steel with no apparent purpose at the rear of the tranny or at the front of the differential above the pinion, this is a dynamic vibration absorber used to solve some vibration problem. If you leave these off, you probably won't break anything but you might not like the car much for the street. Many FWD cars with manual trannys have an absorber in the center of the RH drive axle to suppress torsional vibration.
My first car was a '49 Plymouth with 216cid flat head 6. I wanted a Hot Rod so I bought a 264cid Chrysler 6, rebuilt it and added a cam and alum head. Harmonic balancer on this engine was separate from the crank pulley and easily removable. In order to fit the engine in the car, I needed to cut a large hole in the front cross-member for the balancer so I left it off. It was 40 years later that I suddenly realized that the reason I couldn't keep the flywheel tight on this engine was "no harmonic balancer"