There are many different types of servers, and the hardware you will need will vary depending on what type. For the most part (unless you're working in large company), servers are pretty much beefed up desktops. Servers do typically have fault tolerance (i.e. backup systems, RAID, etc) in mind, and should be dedicated to a specific task such as serving out files, web pages, authenticating users, etc.
Just about any computer can be made into a server. The first thing you need to worry about is the Operating System.
If it is a PC, then you can use one of the Windows Server Operating Systems (Win NT Server, Windows Server 2003, etc). Or you can use Linux (basically any variety). Generally, the newer the better, as long as the hardware meets the basic requirements.
If it is a Mac, you can run OSX Server, or Linux (Yellowdog for PowerPC Macs)
Hardware can vary upon what kind of server. Webservers often have redundant network ports. File servers usually have RAID arrays of disks, and more storage space, etc.
But as thermal said, it really depends on what you're trying to do.
There are special dedicated software programs out there. Winserver 2000 is a good one, it's just pretty much a Windows 2000 but has special options of setting up a broad network. I don't see why you would want to do this though. Lots of work.