You must copy the video url first to paste into a Google Site. You can embed videos from Google Videos, Gogole Docs Video and Youtube.
Then simply Insert > Video > YouTube video. If you want videos from other sources you have to get the embed code from them. i.e.Vevo, Daily Motion etc..
You can embed other videos using
You can produce live streaming and record live using newer technologies such a Google + 'Hangouts on Air' and WebRTC.
Google + 'Hangouts on Air' automatically syncronizes the live video stream to your YouTube account. WebRTC is Peer to Peer and does not require a account/cloud system to connect the audio/video.
WebRTC Video Chat
Insert > Video > YouTube
Paste the URL of your YouTube video