NetHack is an extremely old computer game that can trace its origins back into the eighties with a game called Rogue. I'll refrain from giving you a history lesson, though—you can find the whole thing on the Internet if you care to. It is widely known among geeks as one of the most complex and difficult to learn games ever made, and is much loved by logical minds for the strategy involved in staying alive. NetHack is usually played at a command prompt, with just the symbols you can find on the keyboard, but it's surprisingly easy to interpret and, because of this design choice, it runs on basically every computer with a C compiler. Once you get good at NetHack, it becomes surprisingly easy to win, but that doesn't mean the fun is over, because you can challenge yourself to avoid performing some action (I completed a game without pressing the letter p on the keyboard), get your percentage of won games as high as you can, or set a world record winning the game in about a twentieth of the usual time.
You can play NetHack 'locally', by running it on your own computer, but most involved players prefer to play it on a public server, where you connect to a server that runs the NetHack game for you. This is for several reasons: the server automatically keeps stats and records games for you, it often connects to an IRC channel bot that announces deaths, you don't have to worry about your computer crashing, you can play from anywhere, and you can get the 'bones' of dead players other than yourself.
The most popular server is nethack.alt.org (known as NAO), but there are others such as nethack.eu if your lag is too high. (If you don't live in the US or have a bad internet connection, playing on NAO may be an enormous exercise in patience as your keystrokes take half a second to have any effect.) To play on a public server, you usually just telnet to it. For example, to connect to NAO on my computer, I open a terminal and type 'telnet nethack.alt.org'. If you're on Windows and don't have telnet, check out PuTTY.
If you prefer to play NetHack locally for whatever reason, you can download it from http://www.nethack.org. NetHack is released under a very similar license to the GNU GPL, and you can download the source code if you care to.
The biggest community gathering is the IRC channel #nethack on freenode.net. If you don't have an IRC client, you can join the fun by browsing to http://webchat.freenode.net, choosing a nickname, and selecting the #nethack channel to join. The friendly bot Rodney announces all deaths and wins on NAO and is a convenient portal to statistics and information about the game. There are usually about 200 people on the channel and about 50 people who are semi-active, and people are always happy to answer your questions, however much of a newbie you may be. (If you ask, someone can probably watch your game on NAO and tell you the most likely way to get out of your predicament!)
I'm scorchgeek—if you have any questions about this page or my ttyrec series (see below), feel free to send me a private message if I'm online (/msg scorchgeek <message>), or you can have Rodney deliver a message to me (/msg Rodney !message scorchgeek <message>). In the latter case, I'll get the message when I next log in.
There is also a NetHack Usenet newsgroup, rec.games.roguelike.nethack (link is Google Groups). The volume has dropped in recent years, but there are still people who discuss their games there.
There is a NetHack wiki at http://nethackwiki.com. (There is also a Wikia wiki by nearly the same name, which is beating this wiki in most Google search results, but this wiki was abandoned for the new URL—don't use it anymore, as nobody updates it.)