How does Windows Server differ from any other Windows, say Windows XP?
Does it have an inbuilt server of some kind? I can always start a server on any Windows machine, so I don't understand what Windows Server OS is for?
Windows Server is built on the same codebase as the workstation OS. 2003 is based on XP, 2008 on Vista, 2008 R2 on 7. There are still plenty of differences though.
I'm sure there's many many many more reasons. It could all probably be summed up like this though: If you're going to set up a server, use server grade products, not the same stuff your grandma uses.
One often misunderstood difference is that some versions of 32-bit Windows Server support PAE, allowing use of "all" 4GB or more physical memory. For example, this would allow three "2GB" processes to run "all in RAM" with 6GB of memory. (It would not allow one "6GB" process, because it's still a 32-bit OS. And the "scare quotes" are used because memory usage is not that simple.)
Such support is disabled in all non-Server versions, like XP, because of driver compatibility. Some drivers break with PAE, and consumers would complain. Those running Server would tend to be pickier and "know better".
This is now mostly moot since workstation/consumer versions of 64-bit Windows are common with good driver support, other reasons to require 32-bit Windows are on the wane, and the latest Windows Server (2008 R2) is 64-bit-only.
Jason Berg made excellent points, so I will try not to go in to so much detail on them.
The main differences come down to what they are fundamentally designed to do.
Windows XP, Windows Vista and Windows 7 out of the box are designed to be easy to use for a desktop environment and have many user oriented features.
On the other hand, Windows Server 2003, 2003 R2, 2008 and 2008 R2 are designed purely as servers - they are not designed to look (or sound) pretty, they are just designed so that you can configure and leave it running interrupted - optimised purely for background tasks and services.
There is nothing stopping you from turning off many of the services inside desktop Windows in order to make performance near to the Server, or vice-versa - but it still is not 100% the same.
As for running services and applications on XP - you can always install a third party DNS service or use Apache or other programs - they work very well... However, I am not sure on the licensing constraints of using this edition of Windows for public access - I am guessing it is not allowed, but more than this, if you then wanted to play a game or do some video editing - unless you start to mess around with CPU priories, the server/service may suffer - Server OSes are just designed out the box to serve, and they do it very well.