The Elder Houses were the collected hereditary royal houses of the galaxy. They included the House of Organa and Carise Sindian's family. In ancient times, the Elder Houses were very serious about supporting various monarchies, but at the height of the New Republic, almost nobody took the concept of hereditary nobility seriously any longer. By that time, the Elder Houses had long since become more of a social organization than anything else, and most of its members saw it as nothing more than a genealogical resource and an excuse for the occasional gala.
Most of the Elder Houses shared a number of sacred oaths, one of which was to keep secret anything placed under the sanctity of the royal seal. The most senior members of the Elder Houses formed a ruling council of sorts that could pass certain resolutions. Said council notably had the power to strip a noble from his or her titles and evict him or her from the Elder Houses.
None of the Elder Houses adjudicated succession through strict bloodline inheritance. In other words, any child adopted by nobles and raised as their own was regarded as a genuine aristocrat regardless of their biological parentage. As such, Leia Organa was a legitimate princess of Alderaan, despite the fact that Bail and Breha Organa were only her adopted parents.