Even moreso if you can end the game in one arbitrary move after being signaled to come to a meeting of a secret underworld conspiracy.
Stranger
I’ll be damned! @Stranger_On_A_Train which of the characters in Blitzer will resign by dropping their King? The old immigrant who’s played at the park for 60 years? The young poseur who wears an ascot and affects an English accent? The angsty young woman who carries a copy of Sylvia Plath’s poetry at all times?
As the long-term organizer of an elementary school chess club I can confirm that a shocking number of beginner chess games continue with one (or both) kings in check for any number of moves before either player realizes it.
You’ll have to pay (to watch the film) to find out!
Stranger
It’s not? I’m not a chess player, but I’ve seen it on screen so many times that I assumed it is common.
Usually, you’ll stop the clock and offer your hand to resign. Then, for the sake of the modern electronic boards, the kings are placed on center squares on the color of the winning side so the hardware can recognize the result. Sheets are exchanged and so on.
Of course, you might also offer your hand for a draw, but at top levels, it’s usually clear which is being offered.
I suppose you could knock your own king over in some dramatic gesture, but it’s not really required or done all that often at high level tournaments. Sure, it happens in movies and TV but it’s not like we’ve never seen things exaggerated for dramatic effect.
This was the story I heard/read when growing up and learning to play chess. It’s just a holdover now.
When I was in 4th grade our class had a program where a Soviet expat pro chess player would come in once a week and tutor us in chess. I recall playing a game against a classmate where I somehow managed to capture his king and we had to flag down the tutor to figure out what we were supposed to do now.
I think he wound up declaring our game a draw since we had both screwed up by not realizing his king was in check.
Hi, I’m a retired chess coach.
The answers above have covered the topic (especially Pasta on stalemate), so here are some anecdotes. 
I was playing a grandmaster in an international tournament and announced check. 
He looked briefly at me and the game continued.
Afterwards he explained politely and firmly that you don’t say check - it’s assumed that your opponent has noticed!
Playing blitz chess against another grandmaster, I put him in check. Very short on time, he missed it and put me in check. 
I instantly moved out of check. (doh)
He checked me again … this time I claimed the game as he had made an illegal move in blitz and therefore lost.
In a tournament versus a grandmaster I was easily winning (after 4 hours of the 5 hour session.) I’d won a piece, then exchanged into a simple ending.
I was expecting a resignation, but the game continued for 20 minutes or so.
Then he resigned and explained “I was just waiting for my friend to finish his game so we could go for a meal together!”
It was a key point in The Queen’s Gambit.
We obviously have different definitions of a “simple ending”, if one of them could continue for 20 minutes.
As I said, it was a 5 hour session. We probably played about 5 moves each in that time - but the result was perfectly clear throughout…
It’s a breech of etiquette and in very poor taste to even get to the point where your king would be captured on the very next move. Etiquette dictates that one resign as soon as an untenable position is reached which, unless the victim is a player south of 1,000 ELO points, should be several moves in advance of the coup de gras.
The stroke of fat?
Too much fat can give you a stroke, but I was thinking along the lines of a strong epee thrust to the heart. LOL
coup de grâce
I will of course accept that correction with grace. 
If, however, I decide to choke him to death with a large clump of cheese in order to make it look like an accident, I will revert back to, coup de gras. 
When I was a kid, my sister and I invented a chess variant that we called War Chess. The pieces all moved like in normal chess, but the goal was not to checkmate the opponents king, but to capture all their pieces. The king was just an ordinary piece, and if it was captured, the game would continue. I don’t really remember much about what the game was like, and in any case at the time I had too little experience to really appreciate the difference compared to real chess. I’ve sometimes thought it might be fun to try playing it again.
I suspect that, with reasonably skilled players, that game would end up as a draw, as both sides ended up with a small number of pieces that could just avoid each other indefinitely. If all you care about is keeping one rook or queen alive, it’s awfully difficult to force a capture of it.