An example of an external customer would be a shopper in a supermarket or a diner in a restaurant. These are people who are external to a business as the source of its revenue. They are often the end-user.
The term is contrasted with internal customer, which refers to a person inside a business organization who is reliant upon others inside that organization for services, information, products and so forth. A checkout operator in a supermarket, for instance, will depend on people having stocked the shelves as well as perhaps the availability of customer service representatives. Likewise, waiting staff in a restaurant will rely on the chef's provision of food, so that they can deliver to the external customers, who are the diners.