If you are literally just mentioning a website, as opposed to citing information from it, then no, you wouldn't need to cite it any more than you’d need to cite “London” if you mentioned going for a walk there.
If your essay explores the website (eg. describes its features and sociotechnical affordances), then it’s best to cite it. If your are quoting or taking information from it, then yes, you absolutely should cite it.
Never put a website in quotation marks or italics. A website (eg. YouTube or YouTube.com) stays unitalicised and without marks. The name of particular article or page on a website does get quotation marks.
The only exception to the italics thing is to websites that are the online equivalent of print media, such as newspapers and dictionaries. It’s always Oxford English Dictionary or The Guardian whether it’s the online or print version.