The primary mechanism to copy a geodatabase using geoprocessing is the Copy tool. This tool is found in the Data Management Tools toolbox in the General toolset.
The Copy tool can be used to make a copy of an existing geodatabase of the same type (for example, to make a copy of a file geodatabase). Both the schema and all the contents of the geodatabase will be copied.
You can also use Copy to copy an individual dataset between two geodatabases of any type. For example, you can copy a feature class from an ArcSDE geodatabase to a personal geodatabase.