It's the far-flung future, or an alien planet, or even a Wizarding School. The two characters are talking about the latest plot point over a game of Chess.
No wait... It looks like chess (or checkers or what have you), but it isn't quite the chess we know. It could have multiple, circular, or other odd boards. It could be that the pieces are sentient. It could be something as simple as a different motif for the pieces, or a new unique piece or two. But in some way, the characters are playing a different board game. The characters may or may not still use terms like "Checkmate".
Two of the most common variations are chess with humans for pieces, and layered boards. The "living pieces" version has understandably negative connotations, being a favored trope of villains.
From a narrative and world-building standpoint, this strange, alien board game serves to establish the setting as different from our own in a simple way, while maintaining some verisimilitude by showing the characters do, in fact, play board games. This may be why layered boards are a common choice, as they are visually distinctive (or were, at some point). The game not even be like chess or any other game of ours, because as a visual trope the rules often won't come up anyway.
"Fantastic Chess", "Wizard Chess", and "Future" or "Space Chess" are more specific terms based on setting. Human Chess is a subtrope (so see that page for those examples). Also see Smart People Play Chess. House Rules can act as a Super Trope or a Justification depending on the circumstances.
Very much present in Real Life — The Wikipedia article on Chess Variants puts the number at over 2000 (and that's only counting published ones), with the amount perhaps unlimited. Some of these even enjoy a fair following. Since WWI a subgroup, which includes unusual problems as well as non-standard pieces, known as "Fairy Chess" has been popular, engendering multiple international organizations of players and theorists. See also this page.
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Most Board Game companies have "unsolicited submissions" piles filled with new Chess Variants. At least one of the 5 biggest RPG manufacturers in America automatically rejects them. While they are neat in fiction, and may even be fun to play, they never sell well enough to be worth it unless there is a popular license attached.
Arimaa: Pieces push and pull each other into pits in an effort to get their weakest pieces across the board. Designed to be difficult for computers, but still simple and fun for humans.
Forchess: four-player, two-on-two chess (and it also has a free-for-all version using the same rules).
Tile Chess, where the board is made of multiple free-standing tiles, thus allowing a non-standard board that keeps changing shape, and allows up to SIX players at the same time. It's very fun.
Stealth Chess (not related to the Discworld sort below): The pieces are pictures on tiles, facing away from the other player, and you can arrange the pieces any way you like at the beginning of the game. Since the other player doesn't know what your pieces are you can bluff when moving pieces. They also introduced Super Castling (Rook Teleportation) and some other wacky unique moves.
Blind Chess, wherein the two players play on separate boards, with some sort of divider between them, with only their pieces, and a third (unseen by the other two) person who can see them both maintains a complete view of the game as a referee, telling them when and what they've taken with their latest move, whether moves are legal, and whether they are in check. This is called Kriegspiel (which is German for "War Game.")
Knightmare Chess, where the rules change over the course of the game, based on cards.
Chess960, where the starting position is chosen randomly from 960 possibilities. Invented by Bobby Fischer.
Proteus, where the pieces can change identity repeatedly throughout the game. Also introduces a new piece that can't move but also can't be captured.
Dragon Magazine proposed a game called Dragon Chess, with three boards (representing the surface, the Underdark, and the sky) using miniatures for pieces. Unfortunately they were unable to design a board capable of supporting the weight of the miniatures.
Omega Chess deserves special mention as it gives the impression of actually having the intent to take the place in mainstream board-games on which Chess has historically held a monopoly.
Games like shogi (Japanese chess) and xiangqi (Chinese chess) are related and similar to international chess; both Western/international chess and these games themselves started as variants of the Indian game chaturanga. Some of these games, like makruk (Thai chess) can be played on an international chess board with the same pieces.
There's an entire web site devoted to chess variants called, appropriately enough, Chessvariants.org.
Byzantine chess, which is played on a round board instead of a square one, was created after Pope John VII declared chess the product of "pagans" and banned its play from Christian lands. The monks who invented Byzantine Chess reasoned that, as the circle was a symbol of the sun, and thus a symbol of God, a circular board could not be "pagan". They got away with it.
Alice in Wonderland chess, which uses two boards, one of which is initially empty. Each piece, immediately after its move, "teleports" to the corresponding square of the other board (the move has to be legal before the "teleportation" and you can't move to a square if the corresponding square on the other board is occupied). This creates bizarre situations, e.g. if you want to protect your king from a check by moving a piece in front of the king, the moving piece needs to start its move on the board without the king. This, and other quirks of the game, make traditional defence useless and allows for some attacks impossible in traditional chess (for example, thanks to the fact that the same square on both boards can never be occupied, moving behind enemy lines is much easier).
Guide to Fairy Chess by Anthony Stewart Mackay Dickins (1971). A good history of a particular group of non-standard chess problems, pieces, and boards played since the late 19th Century, and their evolution and additions through the 20th Century.
Martian Chess; each person controls one quadrant of a chessboard, and it uses icehouse pyramids rather than regular chess pieces. Everyone automatically controls pieces in their territory, so every time you capture an enemy piece you lose the capturing piece. The winner is based on value of captured pieces. From those good people over at Looney Labs, it was mentioned in The Empty City as an alternative to the more popular game of Icehouse.
A similar game from the same people is Monochrome Chess; the same idea of a board that has been divided and you only control the pieces in your territory, but it uses regular chess pieces rather than icehouse pyramids. All sorts of recursive game development going on...
And of course the horrifying (if apocryphal) Chess: With THAC0
The page image with the circular board is 3 Man Chess. And yes, there are rules keeping you from taking the rook right next to yours at the beginning, among other things. Some other quirks that arise due to the board include diagonal and horizontal moves sometimes allowing the piece to move in a full circle.
Anime and Manga
In Legend of the Galactic Heroes, there is a variant of chess called Three-Dimensional Chess, which consists on standard chess but with two extra boards above and below the main board, and there are rules for moving the classic chess pieces from one board to the other. Actually makes awesome sense if you analyze it, since, with the advent of space battling, the third dimension has become an important component of military strategy, and this future's version of chess reflects it.
Probably a shout-out to Star Trek, which fits the same description.
In 1963, MAD decided, since old-fashioned chess reflected an obsolete kind of warfare, that the pieces and rules of chess needed to be redesigned to reflect modern military innovation. This "Modern Chess" would be, in other words, Global Thermonuclear War, with bishops, rooks and knights replaced by ICBMs, air raid sirens and fallout shelters, and so on:
Strategy is limited to each player waiting for the other to make the first move. End of game is followed by deathly silence. Unlike old-fashioned chess, there is no winner. There is also no loser. After several years, the radiation subsides enough to permit another game to begin ... if there's anyone left to play it. Also, a new chess set is used which MAD is now designing — with caveman-type pieces.
A particularly irksome moment in the Storm comics has the protagonist teaching a whole city to abandon their bloody gladiatorial games because chess is so much more fun.
One truck commercial shows musclemen playing chess on an outdoor board, using huge marble/stone chess pieces that presumably only they could lift. Yeah.
This commercial for the U.S. Marine Corps, circa 1990 or so. Named one of the 25 most epic ads of all time by Ad Freak.
Futureworld. The protagonists play a game of chess using hologram pieces. The knight is a knight on a horse and a rook is a guy in a small moving castle who shoots arrows at opposing pieces.
The scene in Star Wars, where R2-D2 and Chewie are playing some (holographic?claymation) version of chess, and Han advising R2-D2 to lose...
In Isaac Asimov's novel Pebble in the Sky, set thousands of years in the future, it is mentioned that 3d and other futuristic variants of chess exist, though the game that features as a plot point is of the common sort.
David Eddings' Belgariad occasionally refers to a chess-like game being played by superhuman forces, and the books each have a chess-related title (the protagonist is the titular Pawn of Prophecy).
Tom Corbett, Space Cadet has space chess, played in three dimensions using a cube of intersecting light beams and neutral-buoyancy balloon pieces.
In the Star Trek Expanded Universe novel The Final Reflection, by John M Ford, the protagonist's father studies other races through their chess-equivalents. Of the several mentioned in the novel, klin zha, the Klingon game, is of particular and recurring significance, and has itself several variants.
Gor has its own chess variant, Kaissa. Although it features quite often, the rules are never explained in detail. It serves as a vehicle for sermonising (Tarnsman of Gor), a game played for the hero's life (Assassin of Gor), a means to fraternise with the locals (Marauders of Gor) and others besides.
Gödel, Escher, Bach proposes a game in which (certain parameters of) the rules are represented as pieces on a second chessboard, so that on your turn you can move a piece or change the rules. Or there could be only one board, so that on your turn you are moving a piece and changing the rules. Unsurprisingly, some aficionados have actually tried this out.
A Song of Ice and Fire has cyvasse, a chess-variant with pieces like dragon, elephant, crossbow, trebuchet and mountain. The players align their pieces in a custom starting alignment before the beginning of the game, with a Battleships-style screen dividing the board so your opponent doesn't know your deployment until the game begins. Smart People Play Cyvasse is in full effect.
The Wheel of Time fills this narrative slot with "stones", which appears to be go or a game very much like it. It's stated to be a simplified form of a more complex game called "Sha'rah", whose strategy centers around the capture and/or manipulation of a powerful piece called the Fisher. The children's game Snakes and Foxes (similar to the Real LifeChutes and Ladders, except that it's nearly Unwinnable by Design) is also a major plot point.
The Liavek books have "cylindrical shah". "Regular" shah seems to be chess with a different name and different names for all the pieces; cylindrical shah is a variant in which the players pretend that the board is a cylinder with the two sides touching- meaning that the player can move a piece "off" one side and "onto" the other. There's a complicated incident where some characters get stuck as the pieces in a game of Human Chess between gods, and realize partway through that they're playing cylindrical shah, not the usual version.
The Green Rider series has Intrigue. The game is described as having messengers, generals, knights, spies, and a queen and king. It can be played between two people or with a third member, called the Triad. The Triad can choose to ally with either player but can also choose to be neutral. He or she can also break that alliance at any time. It's intended as a teaching tool for noble youths in the arts of treachery, tactics, and strategy, and is often used as a motif in the stories for the same.
War of the Spider Queen describes a game called "sava", which allows each player to roll dice once per game for a chance to make one of the opponent's pieces capture another. The board is also web-shaped.
Perry Rhodan gives us "Martha-Martha", which is highly popular among the reptilian Tarts (sic) in the Duchy of Krandhor. That the game eventually turns out to be derived from classic Earth chess (introduced by human castaways) even becomes a plot point.
3D chess was used often in Star Trek. Spock was obviously a master. As the rules were concocted after the board was designed just to 'look' futuristic, there are several sets: one involves being able to move several small four-square 'attack boards' with pieces on them to different clipped-on positions on the the three main boards in lieu of a move.
Features as a plot point in the TOS first season episode "Court Martial," where Spock beating the computer at chess is a clue that the computer has been tampered with.
In the 3rd Season episode "Whom Gods Destroy", the sign-countersign Kirk is supposed to use to get beamed back up to the Enterprise — to make sure it was really him and not an impostor — was "Queen to Queen's Level Three". (The countersign was "Queen to King's Level One.")
The Star Trek: The Next Generation episode "Peak Performance" introduces Strategema, which appears to be similar to Go played on a holographic board, in real time and at high speed. Data is unable to defeat a visiting third-level grand master, but is later able to force a draw and cause the Arrogant Kung-Fu Guy to leave in a huff after a game that lasted for only a few seconds but apparently included over 33,000 movies between them. This was some eight years before a similar match in Real Life in which the IBM computer Deep Blue was able to beat (human) champion Gary Kasparov at chess two games to one (with three draws) in a battle of attrition.
Double The Fist had Steve Foxx travel back in time to make the world better by killing Captain Cook, but Cook's crew had force fields and rayguns and fought Steve's allies to a standstill (don't ask) so Steve accepted an offer to settle things peacefully over a game of chess. Holographic 3D chess where moves and captures were declared by announcing Cluedo-style murders.
Lexx had a live-piece chess game in one of the trippy later-season episodes.
One of the things that gets interrupted by the Batphone in the old Batman TV series is a game of chess, played on four or five layered boards.
In the Doctor Who episode "The Wedding Of River Song", The Doctor and an agent of the silence play Live Chess in a gladiator-style pit, with crowds of cheering/screaming spectators.
The Doctor: "The crowd is getting restless, they know the queen is your only legal move. Except you've already moved it twelve times, which means there are now over four million volts running through it. That's why they call it live chess.
There exist a number of chess engines that can handle variant rules range from specific Fairy Chess rulesets to programs like Zillions of Games that allow the definition of a wide variety of variants and boards and attempts to use brute force algorithms to allow the computer to play intelligently.
Archon and its numbered sequel featured chess pieces with varied abilities. Capturing pieces was performed via an arcade sequence.
In Homestuck, every prototyping changes the Variant Chess world known as Skaia and as each player enters it grows more complex
0 Prototypings: A constant stalemate between two kings stuck on a 3 x 3 board
1 Prototyping: Normal chess
2 Prototypings: It becomes a giant CUBE with normal chess rules
Vriska and Doc Scratch seem to enjoy playing this Variant Chess
3 Prototypings: Skaia becomes a planet and the pieces have achieved sentience so it's more like a real time strategy war
4 Prototypings: A giant mass of roots encase Skaia
12 Prototypings: In the troll universe their Skaia looks like a giant ocean planet with with a giant frog named Bilious Slick, the human universe that the trolls created at the center and it's covered in lily pads. It looks like this (spoilers ahead).
Triplicate Girl from the Legion Of Superheroes sometimes splits in three and plays against herself in a game of chess. The chess board is, of course, in two of her colours (orange and purple), and the pieces float above the board. One wonders, though, how she tells the difference between the identical, triangular pieces.