Your configuration object represents your initializations, and is an initialization time component:
"An instance of Configuration allows the application to specify properties and mapping documents to be used when creating a SessionFactory. Usually an application will create a single Configuration, build a single instance of SessionFactory and then instantiate Sessions in threads servicing client requests. The Configuration is meant only as an initialization-time object. SessionFactorys are immutable and do not retain any association back to the Configuration."
The SessionFactory uses the configuration object to create sessions, and sessions represent a period of interaction with the database where a transaction can be started and committed.
"Creates Sessions. Usually an application has a single SessionFactory. Threads servicing client requests obtain Sessions from the factory.
Implementors must be threadsafe.
SessionFactorys are immutable. The behaviour of a SessionFactory is controlled by properties supplied at configuration time. These properties are defined on Environment."