What is aria-label and how should I use it?

It's an attribute designed to help assistive technology (e.g. screen readers) attach a label to an otherwise anonymous HTML element.

So there's the <label> element:

<label for="fmUserName">Your name</label>
<input id="fmUserName">

The <label> explicitly tells the user to type their name into the input box where id="fmUserName".

aria-label does much the same thing, but it's for those cases where it isn't practical or desirable to have a label on screen. Take the MDN example:

<button aria-label="Close" onclick="myDialog.close()">X</button>`

Most people would be able to infer visually that this button will close the dialog. A blind person using assistive technology might just hear "X" read aloud, which doesn't mean much without the visual clues. aria-label explicitly tells them what the button will do.

In the example you give, you're perfectly right, you have to set the title attribute.

If the aria-label is one tool used by assistive technologies (like screen readers), it is not natively supported on browsers and has no effect on them. It won't be of any help to most of the people targetted by the WCAG (except screen reader users), for instance a person with intellectal disabilities.

The "X" is not sufficient enough to give information to the action led by the button (think about someone with no computer knowledge). It might mean "close", "delete", "cancel", "reduce", a strange cross, a doodle, nothing.

Despite the fact that the W3C seems to promote the aria-label rather that the title attribute here: http://www.w3.org/TR/2014/NOTE-WCAG20-TECHS-20140916/ARIA14 in a similar example, you can see that the technology support does not include standard browsers : http://www.w3.org/WAI/WCAG20/Techniques/ua-notes/aria#ARIA14

In fact aria-label, in this exact situation might be used to give more context to an action:

For instance, blind people do not perceive popups like those of us with good vision, it's like a change of context. "Back to the page" will be a more convenient alternative for a screen reader, when "Close" is more significant for someone with no screen reader.

  <button
      aria-label="Back to the page"
      title="Close" onclick="myDialog.close()">X</button>