The Combine server allows the connection of any clients, so you don't need to use the java interface to begin with. Otherwise, it's just a basic java program, so you'd need to look into whatever your OS normally requires to allow the running of such programs.
You need an irc client. The more popular options are:
ChatZilla extension for firefox - https://addons.mozilla.org/en-US/firefox/addon/16
XChat (install from Application -> Add/Remove programs)
Pidgin can handle IRC. It is installed by default.
In the first two cases, install and start the client, then the usual irc commands will work ( /server irc.swcombine.com and then /join #swc-members ). Pidgin needs you to add an account (accounts -> manage accounts -> add) with the server irc.swcombine.com
The Combine Java applet will probably work too. If it doesn't, you might like to install java (I'd recommend using the instructions at http://tombuntu.com/index.php/2008/04/25/10-tips-for-after-you-install-or-upgrade-ubuntu/ point two, and consider all the points as things you might like do).