'We're your nemeses-sis-ses!'
(The Trio, an inept gang of nerds attempting to be supervillains in Buffy the Vampire Slayer, season 6).
Nemesis is a Greek goddess of vengeance and retribution. A daughter of Night, like a lot of Greek deities she's an abstract concept as well as a personified goddess. In addition to being in charge of retribution for wrongs done, she was also responsible for ensuring that human fortunes stayed in balance by punishing people who'd just had a bit too much good luck (balancing out any times Fortune/Tyche had got carried away). It's this last bit that gives the word it's modern meaning. Boiled down to just the concept, without the goddess, Nemesis now usually refers to a person or thing responsible for the downfall of another, and is usually used in a very personal way, with one person being another's personal nemesis.
For this reason, the word is most frequently used in science fiction, comic books and spy thrillers. Because these genres tend to focus on a particular hero gifted with unusal strength/cleverness/gadgets, who takes out normal, puny bad guys ten at a time, the big bad guy who must be defeated at the end of the film is sometimes referred to as, or refers to themselves as, the hero's nemesis. You can see where the word got this usage from clearly enough, from the abstract punishment of someone with too much good fortune to the downfall (except it never is, of course) of someone with excessive gifts or natural/supernatural ability. There's little left of the original Greek idea in the modern usage though, as the idea of the 'nemesis' has become something else, related to the Greek goddess, but not quite the same.
The tenth Star Trek movie, for example, is entitled Nemesis (dir. Stuart Baird, 2002). This is actually a movie I have all sorts of problems with, not least the fact that, brilliant as Tom Hardy is, he looks nothing whatsoever like Patrick Stewart and I'd have trouble believing that they were even related, never mind genetically identical. But the point of the title is that supposedly, in this movie, Picard meets his nemesis. The reason Shinzon is Picard's nemesis is not that he's the strongest or even the most personally affecting villain Picard's ever faced (I would say the Borg Queen in First Contact is both of those) but that he is, essentially, Picard. He is the person Picard might have been if he'd been raised by Remans (as in Romulus and Remus - I've posted on the origins of the Romulans in Star Trek, and I'm sure it's a subject I'll return to again at some point!). One of the most important themes in Nemesis is the question of nature over nurture, and how far someone's personality is due to their genetic (or, in Data's case, electronic) make-up, and Shinzon is the embodiment of this. He's not so much Picard's nemesis as his dark side, or alter ago, or Junigan Shadow, or one of the various multitudinous ways to refer to aspects of a person's personality that are usually hidden and that are often given physical form in science fiction and fantasy. The tragic consequences of his actions bring him more into Nemesis territory, as the general all-round cutsey happiness of Riker and Troi's wedding at the movie's beginning is slowly destroyed, but for the most part this is a more modern development of the nemesis theme, most closely connected with modern genre fiction than Greek myth.
The modern use of the word is not confined to genre fiction, however. Agatha Christie also used Nemesis as the title for one of her Miss Marple mysteries, which I am chiefly familiar with from the Joan Hickson-starrring TV adaptation (dir. David Tucker, 1987). Jason Rafiel, a millionaire, leaves Miss Marple a legacy in his will if she will solve a murder which his son has been accused of. The use of nemesis here is closer to the Greek idea. Rafiel does not know that his son is innocent (and may even have believed him to be guilty) and he calls Miss Marple his nemesis; if she proves his son's guilt, she will be the downfall of his descendants, so she is his nemesis in the modern sense. However, since Rafiel is a millionaire, the whole situtation is also his nemesis - the divine evening out of his good fortune by depriving him of his son, which Miss Marple will complete if she proves his guilt. The Greek sense of the word might also apply to the poor murder victim, who, as it turns out, has been the victim of too much love; love which eventually, in a twisted woman's mind, became fatal (the murderess is played by Margerat Tyzack, who played Antonia in I, Claudius, and who is absolutely brilliant here, cold and chilling and utterly creepy). The girl was too blessed with affection, and divine nemesis stepped in and killed her. I find the Miss Marple mysteries tend to be more emotionally affecting and less Cluedo-y that the Poirots anyway and this one is one of the best - terribly sad, affecting and an intriguing mystery, and the TV adaptation boasts some spectacular performances (as well as Tyzack, Liz Fraser will break your heart as the alcoholic mother of another murdered girl).
The word nemesis will, I'm sure, continue to be used for a number of villains who gain the title simply for being particularly difficult to kill, but I think it is the most powerful when some sense of its Greek meaning is retained - when the 'nemesis' in question avenges some past wrong or puts a serious dent in the hero's good fortune.