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 general, the Apache HTTP server is just a plain old web server designed to serve static web pages. There are plenty of modules which can be installed to enhance Apache's abilities so that it can serve dynamic webpages using various technologies such as PHP, CGI or whatever, but the core of Apache is just a plain old HTTP server.
Tomcat, on the other hand, is specifically designed from the ground-up to serve as a Java Servlet engine. It's primary purpose is to implement the Java Servlet API and execute Java servlets for the purpose of building dynamic websites. Tomcat can also be used as a regular HTTP server that serves static pages, but that is not its primary purpose. (Also, Tomcat is allegedly slower than Apache httpd when it comes to serving static pages.)
The two technologies can be used together through a connector module called mod_jk. This will allow you to use the Apache HTTP server to serve regular static webpages, and the Tomcat Servlet engine to execute servlets.