Hi Sorato,
If you're wondering about the registration options in ArcGIS Server Manager specifically, registering a managed database here means you provide and maintain an enterprise geodatabase into which data will be copied when you publish a feature service to that stand-alone ArcGIS Server site. (In ArcGIS Pro, you need to choose the Copy data option when publishing.)
Publishing to an ArcGIS Server site that has a managed database that you provide is a narrow use case. A managed database is useful when you need to get on-premises data into a geodatabase in the cloud or outside your firewall. When you publish a feature service to an ArcGIS Server site that has a managed database, the data is copied from the source into the managed database. The service accesses the data that was copied to the managed database, not the source data.
Registering a database with the ArcGIS Server site is a far more common workflow. You would do this any time you want the services on your ArcGIS Server site (stand-alone or federated) to access the source data in your enterprise geodatabase or database. The data stays in this database and is not copied when you publish. (In ArcGIS Pro, you need to choose to reference registered data when you publish.) That means people who work with the data through a database connection and those who access the published services are working with the same data.
