Sorry for the quotation marks, but it's not really Flash, but rather an open source version, called Gnash, that you will be 'running.' Yeah, the quotation marks around 'running' are important too. Remember that the Pi is not a very powerful machine, and any Flash heavy page is going to bog it down - especially since most browsers already max out the Raspberry's CPU and RAM. You certainly won't be able to use it to watch Youtube videos. :( If you still need to run Flash, for whatever reason, here's how.
Step 1: Type the following into the terminal:
sudo apt-get install browser-plugin-gnash
Step 2: You will need to restart the browser (if you have it open).
You can now play (very simple small) Flash Games and view annoying advertising. Congrats! :)