Websites like just eat or taste away (ordering food online)..when an order is processed what software enables the restaurant to know exactly what the order was and where it needs to go?

what are you trying to accomplish? are you trying to create a clone? or are you a restaurant owner looking into buying their service?

your question is a bit vague. it's just passing data from a customer to the restaurant. so is "database" the answer you're looking for?

Typically it either adds an order to their website they can check online, emails them, or it sends a fax and calls them.

It's just a shopping cart software, same as any other online retailer, but it's edited/modified for food service.