For youtube specifically I suggest joining the html5 trial. This bypasses flash completely for many videos and so solves the flash issues. Click 'Join the trial' here:

http://www.youtube.com/html5

For other sites that use flash please see below -

A couple steps that help with many Flash issues:

Disable hardware graphics acceleration in Firefox and in Flash

(A) In Firefox, un-check the box here and restart:

orange Firefox button (or Tools menu) > Options > Advanced > General > "Use hardware acceleration when available"

(B) In Flash, see this support article from Adobe: http://helpx.adobe.com/flash-player/kb/video-playback-issues.html#main_Solve_video_playback_issues

Could you also test in Firefox's Safe Mode? That's a standard diagnostic tool to bypass interference by extensions (and some custom settings). More info: Diagnose Firefox issues using Troubleshoot Mode.

You can restart Firefox in Safe Mode using

Help > Restart with Add-ons Disabled

In the dialogue, click "Start in Safe Mode" (not Reset)

Any difference?

You can check for problems with current Flash plugin versions and try these:

• disable a possible RealPlayer Browser Record Plugin extension for Firefox and update the RealPlayer if installed
• disable protected mode in the Flash plugin (Flash 11.3+ on Windows Vista and later)
• disable hardware acceleration in the Flash plugin

• http://kb.mozillazine.org/Flash#Troubleshooting