You can uninstall the game by entering Add or Remove Programs (Windows XP) or Programs (Windows Vista, 7, 8 & 10) in the Control Panel. Once in the list of available programs to uninstall, look for the entry titled ArcheAge and choose Uninstall.
Also, please make sure that you are wiping any extra files that may have remained from a previous installation before trying to reinstall the ArcheAge patcher.
Additional folders can be found under the following locations:
You can also right click on the ArcheAge icon on your desktop, and choose the ‘Open file location’ option. This should lead you right to the ArcheAge installation location. You can delete any files in here to completely remove the game from your system before reinstalling. Be aware that the steps above require you to modify files and folders that are hidden by default. To show hidden hidden files and folders, follow the steps here.