Can I use other people's YouTube videos on my site?
Every time someone uploads a video to YouTube, they know that the video can be embedded on another site (unless they specifically block embedding).
When a video is embedded on a site, this means more exposure for the original creator of the video. YouTubers WANT their videos shared and embedded.
If a site visitor clicks the "YouTube" logo at the bottom right of the video, they will be taken to the YouTube channel of the original creator of the video.