Exposition: The Board Game

Classic games of skill and chance, used to demonstrate characters' personalities.

Battleship, a guessing game originally created for scrap paper "boards" (akin to Tic Tac Toe), but successfully adapted into a commercial form by Milton Bradley. If characters are seen playing this, it usually indicates they've had way too much idle time to kill. Due to the deeply embedded memories of MB's marketing campaigns, nobody ever is depicted destroying a cruiser or carrier, but within 3 turns one player will finally announce "You sank my battleship!" (Particularly suspect, considering a battleship must be hit 4 times before it will sink.) This is more often than not done ironically, or with a lampshade on it, at least recently.
Chess, the supreme Western test of intellect. Expect to see two characters staring at the board for long periods, looking like The Thinker statue. The Spock, The Professor, the riddling villain, and (of course) The Chessmaster will all play this superbly. Show them a game in progress, and they will confidently announce, "Mate in three/five/seventeen." In practice, even the world's best professional chess players would not be able to consistently do this well. Spock, of course, has the excuse that he's an alien. Sometimes, as in House and Robert Heinlein's Sixth Column, it's just a bluff.
- Three-Dimensional Chess. Several varieties of this exist, including one based on the complicated boards seen in Star Trek.
- Frequently seen in movies is a brilliant player who, despite being obviously behind in the game, is able to pull off a masterful combination and win. If he's so good, it makes one wonder how he got behind in the first place.
  - Maybe he started with a handicap, or he let his opponent win at first for the sake of the satisfaction as he finally crushed them.
- And of course, sometimes a character plays Chess with Death.
Go, the supreme Eastern test of intellect. Several orders of magnitude more complex than chess (which is not quite the same as "more complicated than chess"). Knowing how to play well typically signals a character has likewise intellectually surpassed "mere chess". The aura of inscrutable Asian wisdom doesn't hurt either, though in reality playing either game at world championship level is equally difficult (Go took a few years more to be "solved" and computer players be unbeatable, but this is factor of popularity as much as anything else.). And then, in the other direction, there's...
Checkers (or draughts), the archetypical game of casual minds; e.g., young children and leisurely seniors. While definitely a simpler game than chess, checkers may be treated as if it were barely above the level of tic tac toe (noughts and crosses). Extra bathos points for a character using a chess set and board to play checkers. One player can be demonstrated to be far more perceptive or intelligent than the other, possibly even above this game, through them noticing and exploiting a move that allows a triple-jump that ends in the declaration, "King me." Alternately, if it's for humor, they may use their king and just jump EVERYTHING!
Shogi, finally, lies somewhere in between: a Japanese variant of chess, it is typically used in anime as an excuse for old men to sit on porches of rice-paper houses, above the stone lanterns and The Thing That Goes Doink, and discuss in slow grunts the vagaries of life. Unlike the above, Shogi has yet to be "solved" (It's possible to beat a computer) but Hollywood has yet to catch up to this.
Monopoly, a game for the whole family (so long as the whole family understands real estate, mortgages, land development, and math at at least the fifth grade level). Expect lots of squabbling, convenient luck and complicated trades, often extending outside the game.
Scrabble, a game for people with big vocabularies. The Magic Poker Equation applies here. The winner always has just the right letters for a long, high-scoring, but recognisable word, and there's somewhere on the board that it'll fit. They rarely resort to kind of obscure words common in professional Scrabble: aa, cwm, etui. (Although one can occasionally expect Calvin and Hobbes-esque arguments over the legitimacy of such words as "zarf", "Kwyjibo," "jozxyqk" or "zqfmgb.")
Trivial Pursuit, a combination of luck and knowledge. Entire books have detailed not only strategies for both asking and answering questions, but also the game's inaccuracies and ambiguities.
Pictionary, where teams try to guess what the person has drawn. Scenes featuring this will usually have two teams: a) the psychic team, where they're able to guess what their partner is drawing right off the bat, and b) the terrible team, where the encyclopedia-quality drawings of a team member will draw nothing but blank stares.
Dungeons & Dragons While technically a roleplaying game, it is always portrayed using maps and minis. Shorthand for NERRRRRRRRRRDS!!!

There are many more board games, but these are the only ones commonly played.

Often, the trademarked ones will have an in-world Brand X equivalent.

Note: When playing these games, it is vitally important to let the Wookiee win.

Examples of Exposition: The Board Game include:

Fan Works

In Wounded Rose, the very first story of the "Symphony of the Sword" from Undocumented Features, Broken Bird Utena Tenjou ends up joining a Dungeons & Dragons campaign as part of her growing socialization...

...and shortly became renowned in certain circles for being the only player any of them had ever seen who could make a paladin interesting.

In addition to reinforcing that she was not just a jock, it also presaged her future role in the story, as well as a title she earned in a later story, "The Paladin of Titan".

Film - Animated

The Futurama movie Bender's Game includes Dungeons & Dragons as a crucial plot device, in which the main characters end up in a fantasy realm much like the game. The film was already in production at the time Gary Gygax died and debuted later the same year, so it was dedicated in his honor, and included parodies of Dungeons & Dragons-influenced films.

Film - Live Action

Possibly the first appearance of Dungeons & Dragons (or at least something reasonably close to it -- both Runequest and Tunnels and Trolls have been suggested for the actual ruleset used) in film was in 1982's E.T. the Extra-Terrestrial, where it was used to show that Elliot's older brother Mike and his friends weren't necessarily the dumb jocks they might have appeared to be -- as well as to foreshadow a few things from later in the film.
- While the bit of game seen in the film doesn't match D&D exactly, the novelization Did Do the Research and got all the details right in its version of the scene.
The 1993 film Searching for Bobby Fischer is almost archetypal in the way it uses chess as a plot element, a device, and a symbol all at once.

Live Action TV

In the final episode of Freaks and Geeks, entitled "Discos and Dragons", Daniel (James Franco) is forced to join the Audio/Visual Club and the geeks invite him to a game of Dungeons & Dragons. He ends up enjoying it.
In Twin Peaks Diabolical Mastermind Windom Earle is obsessed with chess, and plays a demented "life size" game with real people against Agent Cooper. There are few of the "set piece" elements of Hollywood chess here, but the "supreme test of intellect" aspect is in full play.
The infamous Made for TV Movie Mazes and Monsters uses its eponymous Dungeons & Dragons clone as an extremely heavy handed symbol for all the various Ambiguous Disorders its players suffered from. It's less a cry of "NEEEEERRRRRDS!" than a cry of "CRRRRAAAAAZY PEEEEEEOPLE!" and is responsible for years of bad press for the real game and its players.

Theatre

Chess, of course. The "supreme test of intellect" plays out on many levels at once here, with gambits and countergambits ranging from the individual to the international; and when staged properly, it somehow turns the on-stage chess games into incredibly tense man-to-man struggles while still being essentially two people staring at a game board.