MySQL is type of database. It's massively popular, and is still by far the most widely used free and open source database, but it's all gone a bit corporate lately. Postgres is preferred by many nowadays for political but also technical reasons. OpenStreetMap swapped from using MySQL to Postgres for its main Database a few years ago. For a while The Rails Port code could still work with MySQL, but this is no longer the case (at least not without a lot of hacking)
Of course there's nothing to stop developers still working with OSM data and MySQL.
MySQL OSM tools
TODO: list others