ESHA program versioning follows a convention of x.y.z. Where x is the Program Version, y is the major version, and z is the patch (or update).
With any update to the Program Version (x), or major version (y), there is also an update to the underlying Database. Patches (z) often do not have a database change, but they may. Check the specific Release Notes for any given patch for details.
If the database "lives" on the same machine as the Genesis program itself, the installation of the update itself should update the database automatically. If the database resides elsewhere (or you perform a Client Only -type installation), you may need to manually update the database.
To manually update the database, you can do so with either Database Utility if you've installed it as part of the 'Network' install of Genesis, or SQL Server Management Studio. The above linked documents for either method also include the set of scripts needed.