With the server version of Prince is there just an executable file hosted on the server that any number of client machines can use to install the software on the client or is the software actually installed and run solely on the server with any number of client machines being able to "log in"? If it is run on the server, what kind of hardware is necessary / works best?

The server version is installed on a single server machine and usable by multiple users on that machine, either directly or by accessing it indirectly through a web application or other mechanism. It is not a client/server application as such; what kind of use case for Prince do you have in mind?