Short answers, pulled from the story.
An asteroid family contains asteroids that share specific proper orbital elements including semi-major axis, eccentricity, and orbital inclination. These families are thought to be fragments of past collisions rather than groups with broad characteristics but no common origin.
Japanese astronomer Kiyotsugu Hirayama began his work in 1918 when he first identified the Koronis family. He later found the Eos family and the Themis family while pioneering the estimation of proper elements for asteroids.
Most families form when a parent body shatters during a collision or from large cratering events that do not disrupt the entire parent body. Families have lifetimes of roughly one billion years before smaller asteroids are lost through orbital dissipation caused by Jupiter.
The Vesta family leads with 15,252 identified pieces according to modern counts. The Nysa family reaches 19,073 members total which exceeds the count for Vesta if counting all recognized fragments.
Osculating elements fluctuate on timescales of tens of thousands of years whereas proper elements remain almost constant for millions or billions of years. Modern astronomers analyze proper orbital elements rather than osculating ones to identify more than a hundred families using computer algorithms.