There are two approaches.
The first is to house multiple instances of your app from one instance of Tocmat. To do this, simply rename your war file. Tomcat will deploy the new app within 60 seconds.
The second is to run multiple instances of Tomcat. This is also pretty easy.
Just copy your Tomcat installation and change the ports to something unique in your server.xml file. You will need to change port 8080 and the shutdown port. This approach will allow you to access the different instances by changing the port numbers in the browser's address window as you mentioned in your question.
It's even possible to run multiple instaces of Tomcat from one codebase. Google with 'CATALINA_HOME', 'CATALINA_BASE', and the
word 'multiple' and you will find plenty of information on this.
There are some other alternatives as well but these two are the simplest.