Database email to know the history of backups
There are various SQL scripts available online, for eg: one can be found here to retrieve the backup history and then use sp_send_dbmail to send the email for the retrieved backup history as needed by scheduling it via SQL agent job.

However, the method that i have been using is to make use of Powershell to send such reports as described here Use PowerShell to Report SQL Server Backup Status

You can save the above script and execute it manually whenever you need to. However, to get the most out of it, its recommended scheduling it daily by using SQL Server Agent. Doing this will give you early notification of issues and allow you to act quickly to get those databases backed up.