Etymologically, “mercenary” derives from Latin merces meaning salary or pay, so from this POV a mercenary is someone fighting in a war simply because he gets paid for it and not for any other reasons (having been drafted, idealism, etc.). The Geneva Convnetion defines a mercenary like this:
It does include the restriction that nationals of either party involved in the conflict are not mercenaries, but remember the purpose this definition fulfils. It’s one of those legal “for the poruposes of this statute” definitions that don’t necessarily coincide with what is generally understood under this term. Since the status of captured mercenaries is worse than that of regular soldiers, the Convnetion saw it fit to restrict the mercenary definition that way. I think, however, that it is fair to call someone who is “motivated to take part in the hostilities essentially by the desire for private gain” a mercenary, because that’s the essential part of the word’s meaning, even if there’s more to the purely legal definition.
As for whether mercenaries exist today, I found this press release interesting. It’s from a “private military company,” Northbridge Services, that has repeatedly been called a “mercenary company” in the media.