Thank you for the questions. System databases are defined by Microsoft and are needed for SQL Server to operate. These databases include Master, Model, MSDB, TempDB, Resource, Distribution database used in replication as well as the ReportServer and ReportServerTempDB databases used for Reporting Services.
User defined databases are created by DBAs or Developers for specific applications. These databases can support custom development or third party applications.
In terms of the different types of databases I am not exactly sure what you mean. You can have read-only databases versus read-write. There are databases for line of business needs, reporting, staging for a data warehouse, data warehousing, etc. Please let me know what you are looking for here and I will try to provide additional assistance.