That sounds more like a problem with Adobe Flash Player (which is what is actually used, within Firefox, to view Youtube videos) than with Firefox itself.
To find out if it is, go to your list of installed programs (via the Control Panel and 'Add/Remove Programs', 'Programs & Features', or whatever your version of Windows calls it). Uninstall 'Adobe Flash Player Plug-in'. Restart your computer. (That's important!). Then download and reinstall Flash Player from here:
http://get.adobe.com/flashplayer/
(NB: Deselect the 'Optional Offer'!).
Then restart your computer again. (That's nowhere as important as the first restart but I'd still advise it). With a bit of luck, Youtube might then be OK.
If you do decide to reset Firefox, go to Tools > Extensions > Add-ons first, and note down the names of any extensions that you want to keep, as you'll need to reinstall them after resetting. However most other things (such as your bookmarks) are unaffected by resetting. Details here:
https://support.mozilla.org/en-US/kb/reset-firefox-easily-fix-most-problems