Up to version 12, Opera made their own engine for Opera, called Presto. But in 2013, they decided to re-create the browser from scratch, built on Chromium with the Blink engine. The first version of the "new Opera" was Opera 15, and the "old Opera" was discontinued. But as the new browser was built from scratch, it lost many features. Therefore, there are still many users that want to use Opera 12. As it differs very much, there's a second wiki page for that version. You can access it here: OperaBrowser12.
The best and easiest way to install Opera is by just downloading the deb-package from Opera's website. When you install the package, it will add their repository, automatically configured for updates.
When you have downloaded the file, you can install it by double clicking on the file to open Ubuntu software center. If you don't want to add the repository, you can install the package by opening a terminal in the directory where you have downloaded Opera, and run this command:
sudo dpkg -i <opera-package-name>
When you do this, you will be asked if you want to add their repository or not.
You can read more about this here: http://deb.opera.com/manual.html
You can convert the deb package with this script: https://gist.github.com/ruario/8416e36372f1a976a713
Install this package to get flash:
sudo apt-get install pepperflashplugin-nonfree
Then restart Opera. If it's not working, you may also have to run:
sudo update-pepperflashplugin-nonfree --install
Java is not supported, as they only provide a NPAPI plugin (which are not supported on both Opera and Chrome on Linux).
Get H.264 video working
If you are having trouble with videos on the web that are not flash, it is probably a video that uses the H.264 codec.
Right now Opera doesn't have support for H.264 by default. You can install it by running this command, and then restart Opera:
sudo apt-get install chromium-codecs-ffmpeg-extra
Note! Unfortunately, Opera has some issues with using the library sometimes, because it's so tightly integrated with chromium. There is no 100% good solution of getting H.264 support in Opera right now.
A more detailed explanation of the issue:
Basically Opera are having trouble with the licensing. The library that provides many codecs that Opera uses is called ffmpeg. However, they have removed the H.264 part of it, because even though ffmpeg has a free license, the patent pool of H.264 will demand license fees from commercial projects that are using implementations of it, like Opera.
Chromium and Chrome, however, do have support for H.264 in their ffmpeg library, because Google has payed the license. You can install this library, without the rest of the chromium package, by executing the command above. Then Opera will pick up the library and use it.
You can also read more about this here: https://forums.opera.com/discussion/comment/15223803#Comment_15223803