The phrase that Amazon used when launching lambda was 'deploy code not servers'. To me this sums up what 'serverless' means. It means the developer doesn't have to worry about servers in any way.

With AWS Lambda/API Gateway (and arguably with Google App Engine before it) you take away the toil of having to:

  * Manage/deploy servers
  * Monitor/maintain/upgrade servers
  * Figuring out tools to deploy your app to your server
  * Scaling an app globally.
  * Coping with outages in a data-centre/availability
  * Worry about load-balancing & scaling infrastructure 

So obviously there are still servers there, but they are largely invisible to the developer.

I think this is more than just a marketing gimmick. It is part of a big change in application architectures.

At one end as a small developer of of web/mobile app this can considerably reduce the amount of code/maintenance you need.

At the opposite end of the spectrum the likes of Google (Borg) and Facebook (Tupperware) have developed their own in-house solutions where by servers are largely abstracted out as an entity that developers need to worry about.

Managed docker services (e.g. Google Container Platform, Docker Cloud) are another approach of achieving a largely 'serverless' goal.

(edits for formatting)