When you enter a URL such as http://www.mousetech.com/blog your browser extracts the host/domain name part of the URL (www.mousetech.com) and uses the OS name resolution services to convert it to an IP address (18.104.22.168). The exact mechanisms that are used to do this are usually OS configuration options, although among the most common resolvers are included looking in the /etc/hosts file (or its Windows equivalent) and contacting a Domain Name Server (DNS).
Once your browser has done this conversion, it opens a connection to that IP address. If you have included an explicit port ID such as 8080 as part of your URL, the browser will contact that port. Otherwise, the default port ID is used: 80 for http URLs or 443 for https URLs.
Notice that in all of the above cases, the operative word is browser. Not mention of server, much less of Tomcat.
If you have direct control of the client computer(s), you can override their default behaviors as far as domain names go. In fact, you could add an entry to your /etc/hosts to map www.google.com to 127.0.0.1 (localhost) if you wanted. Although you might regret it.
Port numbers, however, are hardwired into the browser, so short of building a custom edition of the browser (and lots of luck doing that with IE!), your only alternative is to supply a server that listens on port 80.
There are several ways to do that. The major-league solution is to use Apache as a proxy plus mod_jk or mod_proxy. You can also use a simple proxy server, such as squid. In Linux, you can even do port remapping at the OS level (iptables). However, most people either use Apache or they modify Tomcat's server.xml file to listen on port 80 instead of its standard port of 8080.
One caveat, however. Port 80 is protected on a lot of OS's. That means that only privileged (root/admin) applications can use it as a listener port. That's a security risk, since the entire Tomcat system must run privileged, and thus, by extension, all the webapps. That's one reason why Apache is popular as a frontend. Apache can be launched as a privileged user app that then downgrades itself to a less privileged status. Tomcat can't do that.
Being persecuted doesn't in any way prove your righteousness or your beliefs. Many people get persecuted because they are repugnant or annoying. Or just because they can be.
Stinging nettles are edible. But I really want to see you try to eat this tiny ad: