Short answers, pulled from the story.
Orbital inclination is the angle between an object's orbital plane and a reference plane, expressed in degrees. For satellites orbiting a planet, the reference is usually the planet's equatorial plane. For planets in the Solar System, the standard reference is the ecliptic, the plane of Earth's orbit around the Sun.
A critical inclination of 63.4 degrees is the angle at which an artificial satellite orbiting Earth experiences zero apogee drift, meaning the high point of its orbit does not shift over time. This property makes it a practically useful orbital configuration for satellite engineers.
A prograde orbit has an inclination between 0 and 90 degrees, meaning the satellite moves in the same direction as the planet's rotation. A retrograde orbit has an inclination between 90 and 180 degrees, meaning it travels in the opposite direction. An inclination of exactly 90 degrees is a polar orbit, passing over the planet's poles.
The lunar inclination problem refers to the fact that the Moon, though it was once inside the critical orbital distance from Earth, never had an equatorial orbit as standard formation scenarios would predict. Peter Goldreich identified this anomaly in his 1966 paper on the evolution of lunar and planetary moon orbits, and various solutions have been proposed since.
For exoplanets, inclination measures the angle of the orbital plane relative to the plane of the sky as seen from Earth, rather than relative to a planet's equator or the ecliptic. An inclination of 0 degrees means a face-on orbit; 90 degrees means an edge-on orbit, which allows the planet to be seen transiting its star.
Pluto has an inclination of 17 degrees to the ecliptic, Eris reaches 44 degrees, and the large asteroid Pallas sits at 34 degrees. These high inclinations stand in contrast to the modest tilts of the eight major planets and likely reflect a more turbulent formation or dynamical history.