Before you do that either check the md5sum or at least the file size to make sure you actually downloaded the correct file. I've seen problems with downloads from Sun's website before where the full file was not downloaded for some reason.
When you run that file (after giving it run permissions as shown in the previous post) it should ask you to agree to their licensing blah blah and then it unzips itself. From there you can run Java by simply running the java binary under the bin directory in the unzipped java folder.
If you want to install it globally have root move that folder to somewhere such as usr/lib. You will probably want to create a symlink at /usr/lib/java that points to that folder. In addition, you set set the env var JAVA_HOME=/usr/lib/java and add /usr/lib/java/man to your MANPATH if desired. The java bin directory should also be added to your PATH or a java symlink pointed to the java bin directory can be put somewhere that falls under your PATH (such as /usr/local/bin).
Plugins:
all you need to do is make a symlink in your browser plugins directory that points to the appropriate java plugin (/usr/lib/jre1.5.0_12/plugin/i386/ns7/libjavaplugin_oji.so for instance). If you don't want to install the plugins globally then use the browser plugins directory in your browser folder under your home directory (it will be something like .mozilla/firefox/plugins).
If you don't know how to make symbolic links run
and read what it says. Try man command_name for any command you don't understand (command_name --help also can be helpful).
You can google for a good primer on Linux to help get you up to speed. One I like, though it has some distro-specific stuff in it, is the Slack book found at
http://www.slackbook.org/. If you read that you will feel much more confident in managing your system.