Larry, like a lot of other folks, you probably haven't thought about exactly what the words "a server" really mean (to paraphrase a late president).
Hardware vendors use the term "server" to refer to a particular type of configuration which generally is distinguished by high-performance disk drives, options for remote diagnosis and/or control, and possibly even amenities such as hot-swap power supplies. However, in my previous job, I bought these "servers" to act as client machines because the users were pulling down large mainframe files and acting on them locally.
The most common generic use of the term "server" - and I think the one you meant, is LAN server - which refers to software installed on some computer (often a hardware "server" box) that's responsible for sharing out resources over a network. The exact details vary - Novell servers were dedicated machines running a custom OS, nowadays many shops are running Windows servers, and they run basically the same OS as client machines do - just with fewer limitations on resource sharing.
A "server" in the software sense is really any program that waits for client requests and, upon receiving one, serves up some sort of response. A LAN server falls into this category, as do web servers, Java application servers (which often also provide web server capabilities), time servers, mail servers, ftp servers, and many other functions. Software servers are generally lightweight enough that one machine can and often does host multiple software servers of different types and sometimes even multiple servers of the same type (for example, multiple copies of Apache).
Going back to your original question for a moment. To run a java application, you need 2 things. 1) a Java runtime. 2) the java application. Either, neither, or both of these can be places on a LAN server. Just as long as the client machine can access both items - whether through the local filesystem or a LAN share, the Java app will be able to run. The same is actually true of Windows apps as well, except that Windows apps usually call so many outside DLLs and registry entries that the job is next to impossible.
There's also a third type of server I should mention - you can write a Java application AS as server program. I've seen http servers and proxies, ftp servers, and
EJB servers all written in Java. A Java server is just a server application (as defined above) that happens to be written in Java.
Going back to MS Office. At least at one time, I believe that products like Word
were constructed to run off a LAN. However thanks in part to the registry and very much in part to the fact that Microsoft intends to see you pay at least once for every machine you run MS Office apps, you can't just put a copy on a LAN server and use it at will. Instead you'd have to do a separate licensed install for each LAN client.