If you use Windows XP or below navigate to Start > Control Panel > Add or Remove Programs. In the Currently installed programs box, click on the Avira program and then click Remove.
If you have Windows Vista or above you need to go to Start > Control Panel > Programs > Programs and Features, select the Avira application and click on Uninstall.
After that, reboot your computer, run a registry cleaner to make sure that there are no leftover registry keys in your system, download the antivirus again and install it using the serial key received when you purchased. If you still have issues, you can get in touch with the Support Team using the contact information from the official web page.