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 have too much idle time to kill. Due to the deeply embedded memories of MB's marketing campaigns, nobody is ever 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 in a Thinker Pose. The Spock, The Professor, the riddling villain, and 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.
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. 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!
In anime or eastern settings, Mahjong may fulfill the same role as the "less mentally taxing game for leisurely seniors" despite the fact that the game is utterly fiendish. Dominos may also fulfill this role, and if a non Latino creator depicts a Latino area, there will of course be people sitting outside cafés playing Dominos, despite the fact that doing so is more of a Cuban than a universally Latino thing.
In American works, Mahjong is often depicted as being the domain of middle-aged and elderly Jewish women.
Shōgi 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 shishi-odoshi, and discuss in slow grunts the vagaries of life.
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. Also, except someone to flip the table and say Screw This, I'm Outta Here if the game goes on long enough.
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!!! In newer works, or ones with slightly bigger reference pools, Settlers of Catan or another similar “euro-game” can be used to indicate that that the players are NEEEEERRRRDDDDSSSS!!!
