Short answers, pulled from the story.
A geocentric orbit is any path around Earth taken by a natural or artificial object, including the Moon and human-made satellites. By 1997, NASA estimated 2,465 artificial satellite payloads were in geocentric orbits, alongside 6,216 pieces of space debris tracked by the Goddard Space Flight Center.
A satellite in low Earth orbit must travel at about 7.8 kilometers per second horizontally. At that speed, Earth's surface curves away beneath the spacecraft at the same rate it falls, creating a continuous loop around the planet.
A geosynchronous orbit matches Earth's rotation period, completing one circuit every sidereal day (23 hours, 56 minutes, and 4.091 seconds) at an altitude of 35,786 kilometers. A geostationary orbit is a geosynchronous orbit with an inclination of exactly zero degrees, causing the satellite to appear fixed at a single point in the sky from the ground.
The Clarke orbit is named after the writer Arthur C. Clarke. It refers to geostationary orbit at 35,786 kilometers altitude, where a satellite with zero inclination appears motionless in the sky as seen from the ground.
Below about 300 kilometers, orbital decay is rapid and satellite lifetimes shrink to days. At 180 kilometers, only hours remain before the spacecraft vaporizes during re-entry.
The Molniya orbit is a highly elliptical geocentric orbit with an inclination of 63.4 degrees and a period of roughly half a sidereal day, or about 12 hours. A satellite in this orbit spends most of its time near apogee, providing extended coverage over a designated area of Earth.