Short answers, pulled from the story.
The S-IVB was the third stage of the Saturn V and the second stage of the Saturn IB launch vehicle. On lunar missions it fired twice: first to insert the spacecraft into Earth orbit after the second stage cut off, then again to send the crew toward the Moon in a maneuver called translunar injection.
The Douglas Aircraft Company built the S-IVB. NASA administrator T. Keith Glennan awarded Douglas the contract on the 19th of April after eleven companies submitted proposals by a deadline of the 29th of February 1960. Convair came in a close second but was passed over because it was already building the Centaur stage for the Atlas-Centaur rocket.
Douglas built twelve 200-series stages for the Saturn IB, sixteen 500-series stages for the Saturn V, and three test stages. Additional orders for four more 200-series stages and two more 500-series stages were both canceled, the 200-series order in August 1968 before any hardware was assembled.
Starting with Apollo 13, the 500-series S-IVB stages were deliberately crashed into the Moon to generate seismic data for studying the lunar interior. Earlier 500-series stages from Apollo 8 through Apollo 12 were sent into heliocentric orbit. The 200-series stages that launched Skylab crews decayed from low Earth orbit.
S-IVB serial number 212 was converted into the hull of Skylab, the first American space station. It launched on the 14th of May 1973 and reentered the atmosphere on the 11th of July 1979. A second S-IVB, serial number 515, was converted into a backup Skylab but never flew and is preserved at the National Air and Space Museum.
The 200 series was used by the Saturn IB and did not need to restart its engine, so it carried less helium pressurization and used three solid rockets for stage separation. The 500 series, used on the Saturn V, had two Auxiliary Propulsion System modules to settle propellants before engine restart, a flared interstage to match the S-II stage diameter, and only two separation solid rockets.