Code in the client package will get compiled by the GWT compiler down to JavaScript, which will be run in the browser. The bulk of these files are generally cached by default, but you can configure your server to cache these differently if you want. I suggest asking a new question for that though, since that's its own topic.
Code in the server package will run on your server, as Java, and respond to requests from the client.
To learn more about client/server communication, I suggest the official docs. And in general, the Getting Started guide is very helpful for answering the types of questions you're asking.