Short answers, pulled from the story.
A geosynchronous orbit is an Earth-centered orbit whose period matches Earth's rotation: 23 hours, 56 minutes, and 4 seconds. A circular geosynchronous orbit sits at an altitude of 35,786 km above mean sea level, with a semi-major axis of 42,164 km from Earth's center.
Geostationary orbit is a special case of geosynchronous orbit. It is circular, lies exactly in Earth's equatorial plane with zero inclination and zero eccentricity, and keeps a satellite fixed over a single point on the equator. A general geosynchronous orbit can have any inclination or eccentricity, causing the satellite to trace a figure-eight path in the sky over the course of a day.
Herman Potocnik described geosynchronous orbits as useful for space stations in 1929. Arthur C. Clarke popularized their use for broadcast and relay communications satellites in a 1945 paper titled Extra-Terrestrial Relays: Can Rocket Stations Give Worldwide Radio Coverage?, published in Wireless World magazine. The orbit is sometimes called the Clarke Orbit in his honor.
Harold Rosen designed the first geosynchronous satellite while working at Hughes Aircraft, beginning in 1959. The first successful satellite in the series, Syncom 2, reached geosynchronous orbit in 1963. On the 23rd of August 1963, it relayed a phone call between US President John F. Kennedy and Nigerian prime minister Abubakar Tafawa Balewa.
A Tundra orbit is an eccentric geosynchronous orbit inclined at 63.4 degrees that keeps a satellite dwelling over a single high-latitude location for most of its orbital period. Sirius XM Satellite Radio used Tundra orbits to improve signal strength across the northern United States and Canada. At least two satellites are needed to maintain continuous coverage over an area.
When a geosynchronous satellite's thruster fuel is nearly exhausted, operators move it into a graveyard orbit more than 200 km above the geostationary belt. International regulations now require a 90% chance of reaching that altitude at end of life. Deorbiting is not feasible because atmospheric drag is negligible at that altitude and re-entry would consume far more fuel than a simple graveyard maneuver.