Apache Tomcat is used to deploy your Java Servlets and JSPs. So in your Java project you can build your WAR (short for Web ARchive) file, and just drop it in the deploy directory in Tomcat.
So basically Apache is an HTTP Server, serving HTTP. Tomcat is a Servlet and JSP Server serving Java technologies.
Tomcat is a servlet container. A servlet, at the end, is a Java class. JSP files (which are similar to PHP, and older ASP files) are generated into Java code (HttpServlet), which is then compiled to .class files by the server and executed by the Java virtual machine.
In addition to the fine answers above, I think it should be said that Tomcat has it's own HTTP server built into it, and is fully functional at serving static content too. Depending on your java virtual machine configuration it can actually outperform going through traditional connectors in apache such as mod_proxy and mod_jk.
That said a fully optimized Tomcat server should serve static files fast and if you have Java servlets, JSPs and ColdFusion files in addition to static content you may find tomcat does an excellent job by itself.