So, a full stack developer can:
Let's explore those a bit.
There are levels of mastery of all of these things, but that's not really the point. You have junior developers, senior developers and a whole loads in between. But can you build the front-end of a website?
Complicating the issue is the move towards front-end frameworks such as AngularJS, Backbone, Ember and the rest. These raise the bar of front-end web development. Being able to stitch together jQuery borrowed from the web is no longer enough. In the world today, to be considered a good front-end developer you need to know at least one of these application frameworks.
Remember, we're talking a database driven site here. So we need some kind of database, and some kind of server-side programming language.
For the purposes of defining a full stack developer, it doesn't really matter which database or language, so long as you can do it. If you've done it in one language or database, you'll be able to do it in another if you need to. Albeit with a bit of learning.
However, complementing the new front-end frameworks are some server technologies of choice. There's Ruby on Rails of course, but all the cool kids these days are moving towards Node.js.
There's very little point developing a web application if it only exists on your local machine. So to be a full stack developer you need to be able to push your website to a publicly accessible URL.
Again, there are levels. At simple end of the scale you've got Platform as a Service (PaaS) providers such as Heroku or Nodejitsu. These guys do most of the hard work for you with the server, so all you have to do it push your code up and configure a couple of things in your account with them. It really can be that easy.
On the more complicated side of things you could choose to set up your own server, either a dedicated server or a VPS. With one of these you're heading into server admin area, dealing with things like firewalls, ports, routes, security and updates.