What does it mean to say that a web browser/server is "Java enabled"?

For a web browser, saying that it is "Java enabled" means that it supports running Java Applets. That support is either directly within the browser or via the so-called "Java Plug-In".

For a web server, things are bit more ambiguous. The strongest convention is that the web server supports running Java Servlets. That support may be direct in a Java-based web server like Jigsaw or via a connection of one sort or another to a servlet engine such as Resin.