Inability to embed youtube video in Prestashop 1.5 product and CMS pages raised a lot of confusion on prestashop forums and promoted development of various modules.
Fortunatelly, this issue no longer persists in Prestashop 1.6.
1) In backoffice go to the Preferences :: General and set the option Allow iframes on HTML fields to yes . This option is present at least in version PS 188.8.131.52 and higher. You may neet to set the option Allow HTML purifier off as well.
2) Find the video on youtube and click on the Share option. Copy the full code into the clippoard (e.g.
3) In the backoffice, go to the product detail and locate the Description field. Find the Tools button in the editor toolbar, click it to expand and select the Source code button. Paste the youtube code copied in the previous step into the source code pop up window.
4) Press OK to close the source code pop up, press than Save and Stay to actually save the changes. You will see a large questionmark in the place you have copied the video, but this is normal and the video is visible at the frontend.
5) You can use the same approach for CMS pages.