Send email when certain Data is being inserted in MS SQL Server
The quickest solution I can think of would be to create on your specific table an after insert trigger where, based on your conditions (the error no..etc), you'd call the system procedure msdb.dbo.sp_send_dbmail to send a custom email.

You'd have first to configure the mail (create a profile and test it). I don't really think there's a simpler way.

I think that you could create an alert based on a custom error number. But this means that you'll have to modify your current code that inserts data in that table to throw this custom error no, to be able to see the alert.