Java 6 releases can be downloaded on Oracle's website. I think that nowadays you'll have to create an account if you want to download older Java versions; Oracle wants to discourage people using old versions. (You should only use an older version if you have a good reason to do so).
If you only need it to run Java programs (you don't need the Java compiler and other tools), then download JRE 6u26.
I require to install the Java TM SE6 U26 6.0.260.3 browser plugin for Firefox and Chrome,
in order to run an applet, written in an earlier version of java. There are compatibility
issues running the applet using the latest Java TM plugins and I'm not able to update the applet.
I'm not sure of a secure site from where I can download this from?
I do have the plugin installed on another computer - I don't know if it's possible to copy and
install that on to the required host.