Fedora came with OpenJDK Java Runtime Environment, which is a part of Fedora Project. However, many applications still require Sun's Java. When I tried to install FreeMind, a mind mapping software in Java, it required Sun's JRE. I installed the Sun's JRE as following steps successfully.
My environment is Intel Dual Core 32-bit CPU, 4GB Memory, Fedora 12.
Download the latest JDK or JRE at http://java.sun.com/javase/downloads/index.jsp. What I downloaded was jre-6u18-linux-i586.rpm.
Once download you will install the package with: