You may get a banning order from a chat room any of myriad reasons, from spamming to use of bad language. Naturally, the easiest way to avoid the problem of being banned is to not commit any infringements in the first place. But if you do find yourself banned, you can try a few things to remedy the problem.
Read the official code of conduct for the chat room from which you were banned. Understanding why you were banned and what you did wrong is the key to getting the ban overturned. If you can show administrators that you understand the rules and will not offend again, they will be more inclined to overturn the banning order.
Search the website where the chat room is located to find contact details for the website administrators. These are the people who have processed the ban. If you want to get the ban overturned, you must contact them as soon as possible.
Send an e-mail to the administrators explaining your actions. It is possible you were banned due to a misunderstanding, or you may have been banned for a very good reason. Either way, e-mailing the powers-that-be and trying to put forth your case could convince them to retract the banning order.
Enter the chat room using a proxy server. This does not technically constitute being unbanned, but it will allow you to continue using the chat room. Access a site such as Free Proxy Server, YouHide or DaveProxy and type the name of the website into the search bar. It will access the site while hiding your IP address--and most sites ban a particular IP address rather than a user name or e-mail address.