A backing service is any service the app consumes over the network as part of its normal operation. Examples include datastores (such as MySQL or CouchDB), messaging/queueing systems (such as RabbitMQ or Beanstalkd), SMTP services for outbound email (such as Postfix), and caching systems (such as Memcached).
Your code will talk to many services, like a database, a cache, an email service, a queueing system, etc. These should all be referenced by a simple endpoint (URL) and maybe a username and password. They might be running on the same machine, or they might be on a different host, in a different datacenter, or managed by a cloud SaaS company. The point is, your code shouldn’t know the difference.
This is another case where defining your dependencies cleanly keeps your system flexible and each part is abstracted from the complexities of the others…a core tenet of good architecture.
The code for a twelve-factor app makes no distinction between local and third party services. To the app, both are attached resources, accessed via a URL or other locator/credentials stored in the config. A deploy of the twelve-factor app should be able to swap out a local MySQL database with one managed by a third party (such as Amazon RDS) without any changes to the app’s code. Likewise, a local SMTP server could be swapped with a third-party SMTP service (such as Postmark) without code changes. In both cases, only the resource handle in the config needs to change.