I need to check out why a database backup failed in one of our systems.
The log says the following:
BACKUP failed to complete the command BACK
The only information that SQL server itself captures is what you've already quoted:
BACKUP failed to complete the command XXXXXX. Check the backup application log for detailed messages.
This is all that will show up in the SQL error log (SSMS > Management > SQL Server Logs), and the Windows Application errorlog from MSSQLSERVER.
When it says "check the backup application log", it means check the log (if there is one) of whatever application actually performed the backup:
If you can't find it anywhere else, the more detailed error may not have been captured. The SSIS package may not have been configured to do detailed logging, or maybe someone ran a manual backup in the GUI but dismissed or fixed the error on-screen. You may have to find out who attempted to run the backup and ask them why it failed.