If this is for a single user, this is just a fancy configuration for a regular old Mail User Agent (MUA), known to humans as a mail program like Outlook or Mail.app. Fire it up, configure some rules (forward new mail to account, etc) and leave it running.
Based on comments, this is for all users, so you'd probably need to use Exchange Server with server-side forward rules for each user (assuming Windows server platform) or sendmail+.forward files for a UN*X platform, or one of several other alternatives for either. The concept is the same though, a mail server (Mail Transport Agent / MTA) with a fancy configuration.