the easy way: the debian way
In more cases the icedtea plugin is a good solution in order to run java applet in your browser
Sometimes doesn't work, as example with fineco bank, a italian online service.
Sometimes the last version of icedtea-plugin is not up to date; now last version of oracle java is 8.
One easy way to install it, it's explain in the debian's wiki: https://wiki.debian.org/Java/Sun
- download last tar.gz (32 or 64 bit) of java jre
- make a debian package with make-jpkg
- install the .deb package created with dpkg -i
unfortunately when new version of java is released, you must be re-create the deb package, but it takes only 5 minutes
Happy good applets ;)