Short answers, pulled from the story.
The semi-major axis is half of the major axis, running from the center of the ellipse through one focus to the farthest point on the perimeter. It is the longest semidiameter of the ellipse. In orbital mechanics, it is one of the most important measurements for describing a body's orbit.
The semi-major axis is the longer half-diameter of an ellipse, stretching from the center through a focus to the edge. The semi-minor axis is the shorter half-diameter, perpendicular to it, running from the center to the nearest point on the edge. For a circle, both are equal to the radius.
Earth's orbital semi-major axis is 1.00000 astronomical units, where one astronomical unit equals 149.6 million kilometers. The semi-minor axis is 0.99986 astronomical units, a difference of only 0.014 percent, meaning Earth's orbit is nearly circular.
The geocentric semi-major axis of the Moon's orbit is 384,400 km, with a semi-minor axis of 383,800 km given the orbital eccentricity of 0.0549. The barycentric semi-major axis, measured from the Earth-Moon center of mass, is 379,730 km.
For Solar System objects, the orbital period and the semi-major axis are linked by Kepler's third law. All ellipses with the same semi-major axis have the same orbital period, regardless of eccentricity. Newton later derived this relationship from the gravitational constant and the masses of the two bodies.
Not exactly. The semi-major axis equals the average distance only when the average is taken over the eccentric anomaly. Averaging over the true orbital angle gives the semi-minor axis, and the time-averaged distance can differ from the semi-major axis by 50-100 percent depending on the orbit's eccentricity.