I'm assuming this is a browser you've installed? YouTube now uses HTML5 for serving videos by default so Flash isn't used. You could try installing
flashplugin-nonfree but it may not work with your chosen browser. Also, keep in mind that it requires you to run, as root, update-flashplugin-nonfree periodically as it won't automatically update otherwise and it is important to keep Flash up to date.
Edit: My apologies, it seem the browser is based upon Chromium so you'll likely need
pepperflashplugin-nonfree.
