Part of the purview of a rabbi may be acting as a judge, to answer your question.
Back in the day, Moses (Exodus 18:13, Parashat Yitro) sat in judgment, and folks waited in line for him to apply his wisdom to their cases. Seeing the lines, his father-in-law Jethro advised him to delegate his work. So Moses appointed wise men, full of knowledge and understanding, with no greed or ulterior motive, to judge.
Eventually folks set about organizing judges in cities and towns, with banks of three judges for locales with 120 or fewer inhabitants, and 23 judges (a Sanhedrin Ketannah) in the Temple precincts in Jerusalem, and 71 judges sitting in the Temple (the Sanhedrin Gedolah). Over the course of history, judges were appointed by the “laying of hands” (“semikhah”) performed by the person doing the appointing, just as Moses did for Joshua in making him Moses’ successor.
When discussing judges, Maimonides (1138-1204 CE) expounded on (and expanded) the original qualifications, saying judges should be wise and sensible, learned in law, languages, and general knowledge, be acquainted with medicine, math, astronomy, astrology, the ways of sorcerers and idolators. A judge must not be too old, should have children, should not be a eunuch, and on and on.
Those were those days. Nowadays, for the most part our clergy may serve in place of those judges. Clergy, granted semikhah, sit in battei din (plural of beit din), houses of judgment, in groups of three.
What do they judge? They judge matters of divorce, kashrut, conversion. They may supervise the construction of a mikveh. They may certify mohalim, or verify the merit of hekhsherim. In the Orthodox community, they may also sit in judgment on civil cases – viewed as arbitrators outside the secular court system. If their judgment in those cases is based on mutual agreement, it becomes legally enforceable also in secular jurisdictions. If they rule without a tangible agreement, it may be considered mediation and thus may not be as binding in the eyes of the secular court.
In Israel, there is a network of beit din courts operating under the supervision of the Supreme Rabbinical Court in Jerusalem, which has jurisdiction over the Jewish population in all matters of personal status—marriage, divorce, Jewishness, conversion. It operates separately from the secular court, the beit mishpat, though in recent years there have been requests for intervention of the secular Supreme Court over the Rabbinical Court.
One example of the latter is a 2014 divorce case in which a husband had been in a coma for seven years, the Rabbinical Court granted a divorce (a get) based on the presumption that the husband would have agreed to grant the wife the ability to remarry while still young (this is called a get zikui), and senior haredi rabbis denounced the ruling and influenced the Rabbinical Court to review the case. Effectively this was a third-party appeal by someone who had no connection with the couple at all. The legal services organization representing the woman requested the secular Supreme Court to review the practices of the Rabbinical Court allowing such an appeal. The Supreme Court issued an injunction against rehearing the case. (That was in early 2017, and though a similar final ruling was expected, we’ve been unable to find news of it.)
The batei din on which Rabbi Adelson may sit are usually much less controversial than that case, but no less judicial.