The below are few tables provides the essential information about the logshipping databases.
For more information Browse BOL for "Log Shipping Tables and Stored Procedures"
All these tables are stored in MSDB database.
log_shipping_primary_databases -- Displays configuration information for primary databases on the server.
log_shipping_primary_secondaries -- Displays mapping information between primary databases to secondary databases.
log_shipping_secondary -- Displays secondary server database details along with Primary server database information.
log_shipping_secondary_databases -- Displays configuration information for secondary databases on the server.(Run command from Logshipping Secondary Server)
Using the above logshipping tables we can get the last restored log file, last backup file, last copied file, backup source directory, backup destination directory etc.,