These days the best way to do it -- period -- is "yum search jre" or "yum search java". Then type "yum install" whichever Only if that hasn't worked should you go to the Java home page and find the file with rpm in it to download.
Whenever possible download any package through your distro repository or through repositories which are for your distros. Unlike the people responsible for Windows, we value our packages working together in a stable environment. I'm using a web browser which I installed outside my distro right now. I accept the consequences because my distro, which isn't Fedora, does not supply a browser which meets my needs -- even with all the crashing and refusing to surrender the memory it has taken. I don't have to accept this behavior and normally I wouldn't. You shouldn't have to, especially now Java is available through some Fedora repositories.
EDIT: Let me address ron mexico's answer. To install anything you need root privileges. You get them by either using the old UNIX program su or by using the Debian program sudo. Fedora does have sudo but unlike, say Ubuntu it is not enabled by default. You do it in the /etc/sudoers file. You have to open a terminal, become root, type visudo and read how to edit it, then add a line in the format :
jplatt ALL=(ALL) ALL
(taken from my /etc/sudoers file).
Fedora uses the rpm package management system. To install packages you don't have the binaries for on your system (usually these binaries are in rpm format) you usually use yum. You can do an rpm -ivh http://www.example.com/Fedora/RPMS/example-2.6.28.rpm (I just made up that example) and it will download and install example, but yum install example is much easier. The relationship is the same as the dpkg package manager and apt in Debian and Ubuntu. For most third party programs like Java you can also download the source and install it or run bin files by changing the permissions to executables (chmod +x example.bin) then running the file directly (period slash tells the shell to look in the current directory for a file so it would be ./example.bin). But again I can guarantee that if you use yum it will run much more smoothly.