If You Must Use Flash
Generally speaking, my understanding is that Adobe no longer supports flash in Linux - but that the Google Chrome project has decided to integrate and support flash on their own. So Google Chrome would be the only browser that you can use at the moment out of the box, so to speak.
I have managed to get flash to work on other browsers from a recent install on an older 32 bit machine by using a flash-sse plugin (Shockwave Flash 11.1 r102). This build is made for chips that do not have SSE support. I don't know what machine you are using but this is important information.
This was not on an Ubuntu machine but an Arch Linux box running Gnome, but you should be able to find an older version of flash or flash-sse for Ubuntu.
What will happen is that the newer versions of Firefox (I have v27) will block older flash versions automatically with a security warning. You will have to click Activate ... and then Allow ... to view flash. Personally I prefer this as I don't like flash adverts streaming to my machine and using bandwidth.
The security risk, if any, can be mitigated somewhat by sending file locations to &>/dev/null, so that all data streaming in will be shredded on arrival.
[update] I have just installed Chromium v32+ and the above setup (Shockwave Flash 11.1 r102) works for Chromium too - without the grumbling that Firefox exhibits.
For Ubuntu you might be able to find a Debian package that will work. A good link to follow is https://wiki.debian.org/FlashPlayer. It seems that Flash 11 is the last version to work on Linux.
Otherwise Use HTML5
Alternatively, you can enable HTML5 instead of flash on any HTML5 capable browser (especially Chromium) by following this link https://www.youtube.com/html5 and clicking Request the HTML5 Player. From this point forward your Youtube videos and some others will play on any HTML5 capable browser, but I found on my older machine this was too resource intensive. On newer machines I don't think this will be a problem. This is where all streaming content is headed anyway and is why Adobe Flash will be phased out in the longer term.
Best Answer
The Flash Player can be installed from the Multiverse repository:
sudo apt-get install flashplugin-installer. You need to make sure that you have the Multiverse repo enabled: System Settings > Software & Updates.An easy and quick way to get it working though is to just install Google Chrome by going to this link and following the directions: https://www.google.com/intl/en/chrome/browser/
Chrome builds Flash into the browser itself so it means if you have the browser, you have Flash.