From OpenStreetMap Wiki
Search engine services
These options are available as services you can call
- Nominatim is deployed on OpenStreetMap servers, and powers the search on the front page as well as offering an API.
- MapQuest also run nominatim with OSM data as part of their open initiative
- OpenCage  searches across OpenStreetMap, geonames, geoplanet among other sources
- YaCy - a P2P open source search engine
Search engine software
These options can be deployed to your own server.