First of all, it's not a "directory" name, it's a context name. As Vijatha has noted, Tomcat can host multiple applications at one time, and the context part of the URL indicates which application gets to process the URL request. The fact that Tomcat will by default take a war file and explode it into a directory with the same name as the WAR under the TOMCAT_HOME/webapps directory is purely an implementation detail not actually covered by the
J2EE spec.
If you want an application to be at the root of a URL without a context name, you have to define it as the sole application within a virtual host. Tomcat allows setting up multiple virtual hosts, but it's not a minor task and from the sound of it, you don't have the rights to make the needed Tomcat configuration modifications in server.xml
More often, people front Tomcat with the Apache httpd server (or something like IIS) and set up a virtual host there that pipelines requests to the desired Tomcat URL context.
The secret of how to be miserable is to constantly expect things are going to happen the way that they are "supposed" to happen.
You can have faith, which carries the understanding that you may be disappointed. Then there's being a willfully-blind idiot, which virtually guarantees it.