Remember just because something is available on the internet does not mean it’s free to use, most content will be subject to copyright protection until it expires (70 years after the death of the author). Most websites contain a copyright statement or terms of use setting out if and how their content can be reused.
If you want to use a YouTube video in your teaching you should consider
In determining whether a video is legitimate you should consider whether the video has been uploaded by the rightsholder or with the rightsholder's permission.
For example a clip from a BBC programme which appears on the official BBC studio YouTube channel would be legitimate. A BBC programme uploaded by YouTube user joebloggs1 is more likely to be infringing. We would not encourage showing potentially infringing content during a lesson or distributing links to such content to students which could be considered secondary infringement.