A long standing issue for users of 64-bit Linux-based Operating Systems has been a proper lack of 64-bit support for Java in web browsers. Sun Microsystems (now Oracle) had added proper support for this with the release of JavaSE 6u12, however, there is still a bit of work to be done out of the box.
The path of the web plugin has moved, and now resides at /lib/amd64/libnpjp2.so. Luckily, it's really as easy as creating a symlink to this file after Java has been installed.
For example, on my Fedora system, I installed Java 1.6.0_25 and it ends up being installed to /usr/java/jre1.6.0_25, and I am using Firefox 4 as my web browser. So I run the following command:
$ cd ~/.mozilla/plugins $ ln -s /usr/java/jre1.6.0_25/lib/amd64/libnpjp2.so libjavaplugin.so
And that's it! Restart the web browser and everything should be working properly.