As others pointed out, the most straightforward way to deploy a WAR is to copy it to the webapps of the Tomcat install. Another option would be to use the manager application if it is installed (this is not always the case), if it’s properly configured (i.e. if you have the credentials of a user assigned to the appropriate group) and if it you can access it over an insecure network like Internet (but this is very unlikely and you didn’t mention any VPN access). So this leaves you with the webappdirectory.
Now, if Tomcat is installed and running on bilgin.ath.cx (as this is the machine where you uploaded the files), I noticed that Apache is listening to port 80 on that machien so I would bet that Tomcat is not directly exposed and that requests have to go through Apache. In that case, I think that deploying a new webapp and making it visible to the Internet will involve the edit of Apache configuration files (mod_jk?, mod_proxy?). You should either give us more details or discuss this with your hosting provider.
Update: As expected, the bilgin.ath.cx is using Apache Tomcat + Apache HTTPD + mod_jk. The configuration usually involves two files: the worker.properties file to configure the workers and the httpd.conf for Apache. Now, without seeing the current configuration, it’s not easy to give a definitive answer but, basically, you may have to add a JkMount directive in Apache httpd.conf for your new webapp1. Refer to the mod_jk documentation, it has a simple configuration example. Note that modifying httpd.conf will require access to (obviously) and proper rights and that you’ll have to restart Apache after the modifications.
1 I don’t think you’ll need to define a new worker if you are deploying to an already used Tomcat instance, especially if this sounds like Chinese for you 🙂