A server is basically a computer running a (usually specialized) operating system and required sofware to "serve" content like websites, media-streams, storage space, etc. The computer can be a regular desktop computer running the required software, or a computer specialized for its purpose, like a web-server or file-server. Usually, servers run a specialized server operating system (there are specialized versions of linux and windows). These special operating system do not contain any graphical programs, since servers often don't have a display because they are managed over the network. To manage the server locally or over the network, the only thing that is needed is a text console. Connecting to the remote-server over SSH for example gives full control and it's like you are sitting right in front of the server.
The two main terms which are used when talking about internet or other networks are client and server. However, both words are sometimes used to describe only the server/client-sofware and sometimes they are used to describe the whole setup of hardware and software.The server is usually the passive component. A server waits until a connection or a request for something comes in. The client is the part which decides what it needs from the server, connects and sends a request to the server and then recieves the requested data.
The server and client software can either run on different computers or on the same computer. When you are surfing the internet, the server can be located somewhere around the world and your internet browser is the client which requests data (the website or a other file to download) from the server. The server-setup described on this website is somehow a mix between server and client. On one hand, it will act as a server by running a FTP-server, maybe a webserver and provide storage space. On the other hand, client applications will add useful functionality, like running a wuala-client (which in this case is also a server somehow :p) or managing downloads with a download manager.
The setup which is often called a homeserver is basically the same like any other server. It is some hardware and software, but it is not as powerful as a webserver serving hundreds of websites or a big downloadserver you can find in the internet. One possibility of a homeserver is a NAS, or you can also use any computer you have at home and set it up as homeserver. This website basically describes the use of a Plugcomputer (a very small computer with enough power to run a small server) and the required steps, or my ideas, how to set it up.