Just precluding some questions:
IRC, or Internet Relay Chat, is a client-server based system, and servers are for the most part independant. For example, if you connect to irc.freenode.net and join #paradice you'll find it to either be empty or full of people talking about some completely unrelated thing; it has nothing to do with the #paradice on chat.psionics.net.
To use IRC:
Find an IRC client; there are quite a few out there, for every platform.
Here're some of my favorites:
ChatZilla - an extension for Firefox and Mozilla (I think Mozilla comes with it, not sure though). With it installed, you'll be able to click on links like [url=irc://chat.psionics.net/paradice]irc://chat.psionics.net/paradice[/url] to get on IRC, and it can also be used independantly, without a link. It runs on anything that firefox runs on, as far as I know.
XChat - My personal favorite; a bit harder to learn to use if you're unfamilliar with IRC, and originally designed for *nix (Linux, Unix, and other misc. POSIX-compliant) OS's, it can be a little odd feeling to someone not used to such things, but it's very powerful, and runs on most mainstream OS's.
Also popular client, although not one I like, is
mIRC, a nonfree windows-only program.
Useful commands found on most clients (Note: many clients let you do // if you actually want to start a line with / that's not a command):
/me <string> - Executes an emote with the supplied string. For example, entering "/me dances" would display
* Ralith dances.
/join <channel> - Joins the specified channel. All channels begin with #. For example, entering "/join #paradice" after connecting to chat.sionics.net would take me to our channel.
/nick <new nick> - Changes your nick.
/away <reason> - Sets your status as away, for reason <reason>. Most clients will display your nick greyed out or otherwise marked in the userlist while you're away.
/msg <nick> <message> - Sends a private message to <nick>. Most clients will open a new tab or window when a reply to one of these is received.
/notice <nick> <message> - Like /msg, except will show wherever <nick> is currently viewing, and does not open a new tab/window.
/whois <nick> - Displays information about the nick supplied. For example, entering "/whois Ralith" would display
* [Ralith] (~
Ralith@soggy202.drizzle.com): Ralith Lupus
* [Ralith] #paradice
* [Ralith] chat.psionics.net :The Internet's Premiere Role-Playing Chat Community
* [Ralith] idle 00:00:17, signon: Thu Nov 03 06:59:07
* [Ralith] End of WHOIS list.
(note that formatting may vary on other clients)
How to register (Registration will prevent others from impersonating you):
Once you've connected, enter "
/nickserv register <pass> <email>". The email MUST be valid, as it will send a one-time-use command to it that you'll have to enter to complete registration. Once you've registered, reconnect and enter "
/pass <pass>" within one minute to log in. If you don't enter it within the time limit, the server will force your name to change to something like "Guest123". Most clients can be configured to automatically enter that after connecting.
If you say someone's full nick, chances are their client will flash to get their attention. Don't abuse this.
[edit: added /away]
[edit2: added note about beeping]
[edit3: added /msg and /notice]
[edit4: minor reformatting]