When writing your magnum opus, you might want to put emphasis on certain facts, names, and so on. There are three good ways and one bad way of doing this.

Let us take a look at the bad way first: Underlining. Depending on the font you use, it might look very irregular. For web content, it might confuse your readers because they expect a hyperlink. Also, it is slightly hard to read.

This is a remnant from the age of typewriters where other text decorations were impossible to do. Let this remnant enjoy its well-deserved retirement. In modern times, we can do without it. It evokes, at least for me, the distinct style of yellow press publication.

So, what are the three good ways, then? To wit, italics, bold, or small caps. Italics are easily achieved in LaTeX with \textit{Text} (or \emph{Text}, which is preferable for a number of reasons; see below) or in HTML with <em>Text</em>. Similarly, you obtain a bold decoration with \textbf{Text} in LaTeX or <strong>Text</strong> in HTML—although <strong> only defaults to bold text. How it is actually rendered depends on the CSS. Last, small caps are capital characters that are set similar to surrounding lower case letters, making them not as conspicuous as upper-case characters. Whether they look good or not depends very much on the font you use. In LaTeX, you can get them using \textsc{Text}, in HTML you have to add font-variant:small-caps; to the current style.

This is pretty straightforward. The one thing you should avoid is combining different modes of emphasis. Bold and italics do not mix well in most cases.

As a parting thought, some words on \emph in LaTeX: You should rather use \emph than \textit because the former command is somewhat smart and can recognize nesting. For example, if your personal style is to typeset figure captions in italics but you need to emphasize something in the caption, \emph will detect this and render the text upright. When using \textit, nothing would happen here because, well, the text is already in italics. If this does not convince you, just think about the flexibility offered by \emph: You could redefine it to work differently for some sections in your document if that strikes your fancy.

To end with a happy note for the poor underlined text: You can use underlining to mark corrections in a text. This might be especially helpful when collaborating on a document. But do not use underlines for anything that is going to be included in the final version of a document.

TLDR: Do not use underlining for emphasis in a text. Use italics, bold, or small caps, depending on your font and your fancy.