In systems design, code that runs on the user's computer is called client-side, and code that runs on a server on the internet somewhere is called server-side. Applications like single-player games can be entirely client-side, and scheduled tasks like indexing files for search can be entirely server-side, but the majority of applications that live on the internet trade off some of the work between the client and the server. These are client-server applications.
The great thing about client-server applications, and one of the strengths of the internet, is that as long as you agree on a specification for client-server communication, you can use any technology on the client side and any technology on the server side. That's how you've seen web sites transition into web services. What at one time was simply a site for sharing photos, presenting your browser with static pages of photos, is now a photo-sharing service, which you can interface with in a mobile phone app, another web site, or a stand-alone application. As a client-side developer, you use the APIs that the service exposes (and some expose the same services in multiple ways to be even more flexible) to take advantage of that service.
ActionScript 3.0 is a client-side technology: it runs in Flash Player on the end-user's computer. And, like a good client-side ...