"What is this application/plugin/widget even called"
It's called HTML.
Seriously, this is a pre-existing feature of any blogging tool. You just write out the
<a> element, with the url for the link as the value of the href (Hyperlink REFerence) attribute, and the link text between the opening and closing tags:
<a href="url/for/link.html">Click me</a>
More on links with HTML:
To simplify this, when using the *advanced* or rich visual editor you can select a bit of text already typed in the textarea, click the format button that looks like a chain link (the 'link' button), and enter the url. In the basic editor, the 'link' quicktag button works the same.