It's an abstract and complicated issue whether your site is a copyright violation. If you're trying to compete with youtube, then it probably is. If you are simply trying to create an additional website using videos from youtube, it probably is not. (However, for the reasons above I would suggest confirming all of this-- it would be a big waste of time to make a website only to have it shut down due to youtube blocking/suing you.) To be clear, I'm not offering any legal advice/opinions. All I'm concerned with is whether the topic is appropriate for discussion here.
Here are the two issues I see (and are relevant to your original question):
1. You should not create a website that is in competition with youtube while using videos from youtube, embedded in your own site as if they are your videos.
2. Specifically, branding is important. I'm almost certain that part of the requirements for using embedded youtube videos is that it is made clear that they are, in fact, videos from youtube (this is a sort of advertising for them).
If you want to add content around the videos, that's fine. If you want to hide where they come from, that's a problem.