What is the least amount of privileges (Administrator, db_owner, etc ) for a user to perform Backup and Restore in SQL Server?
For backups the db_backupoperator fixed database role is the least amount of privileges needed. 

https://msdn.microsoft.com/en-gb/library/ms187510.aspx

For Restores, If the database being restored does not exist, the user must have CREATE DATABASE permissions to be able to execute RESTORE.

https://msdn.microsoft.com/en-gb/library/ms186858.aspx