I want just the text to show up as a link, no URL. I've tried the basic HTML
<a href="www.bing.com">Check out this new website I found</a>
but it doesn't work.
The same syntax also works in questions. See this meta.SO feature request for the original request.
Also, you can always click the 'help' link at the bottom-right of the comment box to bring up this text:
Comments use mini-Markdown formatting: [link](http://example.com) _italic_ **bold** `code`. The post author will automatically be notified of your comment. To notify a previous commenter, mention their user name: @peter or @PeterSmith will both work.
See also this question for known bugs.