By magic. I really want to know why some people want to have their accounts deleted, but well...
Option one: Email Justin at support@armorgames.com from the email you currently have associated to your armorgames account. As him to remove your accounts. This will lead your account to be permanently banned, and you will no longer have access to it. The name will no longer be available.
Option two: Go to your control panel and change some settings. Make sure you set it to no longer wanting to receive news from AG, you don't want notifications for comments and merits. Close your comments. Go to your about and remove all text there, or alternately, leave a message explaining you have left the site. Now go to the email and put in something random like "Iliekfeet45215@feetclub.com" and move to the password field. Change the password to something that contains both letters and numbers, capital and regular, and preferably looks like you smashed your hand or face against the keyboard several times. The longer, the better. Save these changes and log out. Your account is now no longer accessible to you. The name is no longer available.
Obviously, the second option is instant, precise and doesn't mean you have to wait for a busy admin to help you out. The first option will make you wait for a busy admin to come back to you. I recommend the second option.
Which would be a fairly idiotic move when you could take the proper way out, with pride. And with the ability to get back onto the site without having your *** banned for circumventing your ban.