Having correctly installed the JRE you can now pass on and install the servlet container. Servlet container, or web container, is the component server that interacts with the servlets. It is responsible for managing the lifecycle of servlets, mapping a URL to a particular servlet, and ensuring access security. It should implement Java servlet and JavaServer Pages technologies.
As for JRE, you have a few choices here; a brief list is at http://en.wikipedia.org/wiki/Web_container.
Apache Tomcat, GlassFish , and JBoss are most popular and are all available in an open source edition. You may wonder which one is the best choice for running GeoServer. In a production environment, usually the same container is shared among several web applications. You are not going to choose the container; the architects and system administrators made their choices and you have to conform to them. As a beginner, you have the opportunity of selecting it! Apache Tomcat should be your first...