ahh firefox can't install plugins instelf on any OS ive used it in. the problem with the file is you download an RPM, which is for fedora and/or openSUSE/slackware, while linux mint, ubuntu, and debian based OS's use .deb files. the easiest way to install java would be to open a terminal and type this:
"sudo apt-get install icedtea6-plugin"
this may give you an error talking about some missing dependancies, if it does type
"sudo apt-get -f install"
which is what it should say in the error. let that finish, and do
"sudo apt-get install icedtea6-plugin"
again. let it finish, and you should have java working in your browser. i just did this myself 3-4 minets ago on linux mint, so i can guarentee you it works. enjoy =]
-- remember not to include the quotes, and to type your root password when it asks, it woln't show up but it is being typed.
EDIT: try entering this into the terminal, and use it to set your root password:
"sudo passwd"
i know it has sudo in it, but its what you use to set a new root (sudo) password. if it doesn't work then you might have already set one when you installed linux mint, which means you have to either remember it or reinstall and write it down when you set it. you have to remember this password because you will use it allot in linux.