How to embed YouTube video in your Joomla! site
You can of course embed YouTube video in Joomla! articles howeve, iframe tags will be deleted by default when you paste it in Source Editor. You have to disable this feature. All you have to do is to go to Plug-in Manager, click "Editor - TinyMCE", and delete "iframe" from the list of prohibited elements.
If you are not sure, I made a video explaining this so please take a look.