If I were to display videos on my website, how would I go about hiding the videos link?

You can't.

Attempts to hide the link from the user also hide it from the browser. The user can read the source code for any web page with View / Source.

Why do you want to hide the link? Sounds selfish to me. The entire purpose of the Internet is openness, not secrecy.

Rather than hide the link, Can I prevent it being downloaded from my web server? Is there a way to allow it to stream without downloading to the users pc?

Not that I know of... you could attempt to use a streaming media server or a PHP no cache header, but it will still leave a temporary file of sorts that can be viewed with a bit of modification... I'll look into this and see if I can turn anything up.

Use display:none attribute of style.