Nominatim is geocoding software that searches OpenStreetMap data by name and address and, for a given location, looks up the matching address (reverse geocoding). Organisations use it for address search in map applications and web forms. It is delivered as server software and as a Python library that you run on your own server with your own OpenStreetMap import; according to the project site, the official OpenStreetMap website runs on it too.
Nominatim is developed by a community without a legal entity; according to the imprint on the project website, its lead maintainer is based in Germany, and the site names a company as a supporter of parts of the development. The Python code is licensed under the GPL (version 3 or later); a few other parts are under Apache-2.0 or GPL-2.0. When you host it yourself, search queries stay with you; the public instance is governed by a usage policy of the OpenStreetMap Foundation. Development was co-funded through the EU's NGI Zero programme, run by NLnet.