The Union was founded at Brussels in 1919, in the round of international scientific organisation that followed the First World War, and its membership is of two kinds: national members, which are countries adhering through an academy or a national committee, and individual members, who are professional astronomers. Its business is done in divisions, commissions and working groups, and its General Assembly meets every three years.
Naming is the part of its work that reaches the public. The Union is the recognised authority for the designation of celestial bodies and of surface features on them, so the names of craters, of asteroids, of features on Pluto and of exoplanets are settled through its working groups against published rules about what sorts of names are admissible for what sorts of objects. The Working Group for Planetary System Nomenclature maintains the gazetteer.
Its most publicly argued decision was the resolution of 2006 defining a planet in the solar system, which established three criteria and, in consequence, moved Pluto into a new category of dwarf planet. The reaction to that was a useful demonstration of something the Union does not otherwise have to think about: a naming convention agreed by specialists can turn out to be a matter of general feeling, because the public had learned the old list at school and regarded it as a fact rather than as a definition.