An Internet Chess Server is a server out on the Internet where you can log in and play chess with people from all over the world. You can also play against computers, watch other users play, or just hang out and chat.
Many of the chess servers on the Internet today are distant descendants of the original ICS written by Michael Moore and Richard Nash in 1992. Today's ICSes run completely rewritten code, but they have kept enough compatibility that many chess clients still work with all of them. In particular, you can use XBoard and WinBoard as graphical user interfaces to all ICS-compatible servers, and you can use Zippy to hook up certain chess engines to them as automatic players.
Some popular ICS-compatible servers are:
I'm registered as mann on ICC, FICS, and a few other chess servers. I'm no longer a system administrator on any of the servers, due to lack of time.
There are now many newer chess servers on the network that are not compatible with ICS, such as Yahoo Chess, MSN Gaming Zone, World Chess Network, etc., but I don't use them and XBoard and WinBoard don't work with them, so I'm not interested in keeping a list of them on my Web page.
Sometimes people tell me they would like to start their own Internet chess server and ask me for help. I don't have time to help with such projects, but I can give some advice and pointers here.
First the advice: Don't bother. You probably won't get much satisfaction out of starting your own ICS. To be useful, an ICS needs a large community of players so that there are always plenty of people of all strengths available to play. For that reason, players have gravitated to a few existing servers. Drawing a critical mass of players to a new server is difficult at best.
However, if you still really want to run your own server, there are some ICS-compatible servers you can download. One good place to look is the Chessd project on Sourceforge. Another is Andrew Tridgell's version of chessd. Yet another is the old version of the FICS server on the freechess.org FTP site; however, as I understand it, Chessd is based on this code and has fixed many bugs in it, so you are probably better off not bothering with it.
Do not ask me questions about any of this code; it is not mine and I don't have any recent experience using it.