There are a variety of methods to banning someone from a website, forum, game or chat. Some of the common methods and solutions are listed blow. In general your best option to remove a ban is to humbly apologize for your actions which lead to you being banned in the first place.
Banned by IP address
If you are banned only via your IP address you can go ahead and attempt to change your IP address. Make sure to clear your cookies first.
Banned by cookie
Using your browser clear your cookies.
Banned by unique token
Many online games have a unique token that is passed during online gaming. This token is often related directly or indirectly to the CD (install) key included with the game when it's purchased. Only by uninstalling and reinstalling with a new CD key will get you around being banned. This type of ban may also include websites that require specific plug-in applications such as Flash. You would want to uninstall any of these plug-ins and reinstall them. This *may* allow you to get passed this type of ban.
Banned by common information
Some websites will ban other accounts (or new accounts) with related personal information such as email address, passwords, credit card number, physical address, etc. When eBay shuts down accounts they seem to shut down any other accounts with related information. The only way around such a ban is to create a new account with *none* of the same information.
Banned by multiple above
Some methods of banning may include one or more of the above so we'd suggest that you do as much as is reasonable from the above solutions.