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It's probably similar to Office 365 and the email alias option. With Office 365 you can set up a number of email aliases that can all be directed to one inbox. The downside is that you can't send from those alias. You will always only have one "sending" email address, unless of course you pay for another user inbox.

It sounds like you're looking for Google Apps (http://www.google.com/enterprise/apps/business/index.html). Although since they'll let you forward to a free gmail account, it seems you're pretty covered.

I assume that is where Google Apps comes in.

Or a free Gmail account for incoming mail. The only missing part of the jigsaw would be an SMTP server - otherwise your email appears to others with something like 'sent on behalf of' or similar (depending on the mail client).

GoDaddy gives you one email box with your domain name so that when you reply the email, it is sent from the real email account.