Short answers, pulled from the story.
The astronomical unit, symbol au or AU, is a unit of length defined as exactly 149597870700 metres. It was redefined as a fixed conventional unit tied directly to the metre by the International Astronomical Union in 2012.
One astronomical unit is approximately equivalent to 499 light-seconds, slightly more than 8 minutes 19 seconds. The precise light-time for unit distance reported by the IAU in 2009 was 499.0047838061 seconds.
The IAU redefined the astronomical unit in August 2012 because the older 1976 definition did not specify a frame of reference and fully accounting for relativity would have made the definition overly complex. The new definition recognizes that the unit now has reduced importance.
Around 280 BC, Aristarchus measured the Moon-Earth-Sun angle when the Moon was in its first quarter and estimated the Sun was 18 to 20 times farther than the Moon. He estimated the angle at 87 degrees, while the true value is close to 89.853 degrees.
By measuring a transit of Venus from two different locations, astronomers calculated the parallax of Venus and from it the solar parallax. The transits of 1761 and 1769 became an international operation including observations by James Cook and Charles Green from Tahiti, and Jerome Lalande collated the results into a solar parallax of 8.6 arcseconds.
The astronomical unit is used primarily for stellar system scale distances, such as the size of a protostellar disk or the heliocentric distance of an asteroid. It is too small for interstellar distances, where the parsec and light-year are used, and the parsec is itself defined in terms of the astronomical unit.
In the 2nd century CE, Ptolemy estimated the mean distance of the Sun as 1210 times Earth's radius. His figure was approximately correct only because errors in his parallax, his theory of the Moon's orbit, and other factors cancelled out.