This week's book giveaway is in the OCMJEA forum. We're giving away four copies of OCM Java EE 6 Enterprise Architect Exam Guide and have Paul Allen & Joseph Bambara on-line! See this thread for details.
It could be called an application server, but not fully blown application server. It has Servlet/JSP container as someone specified. Inorder to be a full server it still needs EJB container which it does not support. But is more than a web server.
Tomcat is an application server but doesn't suppor ALL of the J2EE stack. Most notibly, EJB is missing.
If your application requires EJB support and you want an open source application server, look at JBoss. It supports the full J2EE stack. It uses Tomcat as it's Servlet conatiner and JSP engine.
In addition Tomcat is fully capable of running as a stand-alone webserver but can also be connected to the Apache Webserver or IIS. This allows you to use the other webserver for handling static files, ASP, PHP, Perl, etc.. and send to Tomcat only those requests that need to be processed with server side Java.