A Full-Stack Web Developer is someone who is able to work on both the frontend or backend portions of an application. Front-end generally refers to the portion of an application the user will see or interact with, and the back-end is the part of the application that handles the logic, database interactions, user authentication, server configuration, etc. Being a Full-Stack Developer doesn’t mean that you have necessarily mastered everything required to work with the front-end or back-end, but it means that you are able to work on both sides and understand what is going on when building an application.
MongoDB -> AWS, HTML5
ExpressJS -> NGINX, SASS
AngularJS -> UX
At the level of a senior full-stack engineer we’re perhaps talking about architecting a modular Backbone/AngularJS front-end whilst optimising content delivery and tweaking hardware accelerated layers in CSS, followed by implementing an async non-blocking backend (which also pre-renders templates on the server), and pushing to an AWS cluster which has been built with security and scaleability in mind. Not to mention design. Designing UI responsively and mobile-first is essential, utilising a CSS pre-processor to save time. And remember to setup Nagios for monitoring. Oh, and ideally when the Continuous Integration server detects a bad build because your end-to-end tests have failed, get it to send you an SMS with the build error messages. Well, the last two are probably for bonus points.
Can you learn both? Yep. I don’t think you should limit yourself if you can help it. Knowing more is never a bad thing. Being good at all, but not a master is an alright place to be in, because you can always dive deeper and learn more, because that is the way of software development.