You can have as much Youtube movies ( that might contain copyrighted material ) on your website as you want. The uploader and Youtube are responsible for the content, not the website that have the embed code on their website. You are not in any danger...
I only can speak for Germany, and according to our laws we arenīt allowed to show or in this particular case embed content that clearly violates copyright, meaning that we can be held responsible for spreading that material thus being sued (well, technically....).
Althought there was a lot of rumour going on about U.S. courts taking sort of a "Fair Use Policy" into consideration.
A very interesting read about that is this article:
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