Tomcat as distributed uses TCP/IP port 8080 for http requests. Web browsers look at port 80 by default, so to chop off the :8080 part of the URL, you'll need to modify the port setting in Tomcat's server.xml.
For Windows, the hostname can be set using the Network settings
applet on the Control Panel. Depending on the OS version and machine configuration, you can also get to this applet off the right mouse button menu of Network Neighborhood.
In commercial-grade server environments, it's common to use Apache as a front-end web server. In that case you wouldn't change Tomcat's port setting, because Apache would be serving port 80. Instead, you'd set up a connector between Apache and Tomcat. That's not as simple a task as it ought to be.
Also, of course, for commercial serving, you'd define the server via DNS.
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.