IRCClient.swf is an irc client written in actionscript. It can easily be embedded into websites. Since it natively speaks the IRC protocol, it does not require you to setup an irc to xml gateway or other silly things.
Java applets must reside on the same domain as the accessed server, thus you have to put a java irc applet on a webserver listening on the same IP address as the irc server. In contrast, the flash irc client can be run on any domain -- the irc server just has to provide a simple service running on port 843, telling the flash player the irc client is allowed to connect to it. Also, flash is more widely deployed than java, effectively making it accessible to everyone -- those who don't have a flash player installed usually know how to use irssi anyway.
The flash player is still in a pre-alpha state; test it here. The source can be retrieved via git:
git clone http://www.hcesperer.org/flashychat/ircclient.git
If you just want to embed a chat into your website, you can use this cgi based chat powered by evility.net instead: