Parlor Games

Before Board Games, Card Games, Tabletop Games, Video Games, and Web Games came along, people just had their own persons to play games with guests. These are known as Parlor Games. In the past, these were used in fiction for the same purpose as Board Games are these days. Nowadays, it's either a Discredited Trope used to show how boring or geeky the people playing are, or it's used as an actual Plot Device.

Simon Says, a children's game where someone gives orders (usually silly things like "clap your hands" or "jump up and down"). Everyone playing has to follow the commands as long as they're preceded by "Simon says". So if "Simon says clap your hands" you have to clap, but just "clap your hands", you don't. You're out if you either follow the command without the "Simon says" or don't follow it when they do say it. One variation has Simon do an action in addition to saying one, but you must do what "Simon says". Usually, Simon will do and say the same thing, but it could lead to situations where "Simon says clap your hands" but he physically jumps up and down as a trick; the proper action is to clap your hands. Last of the group still in usually gets to be the next one to call out the orders. This game can be challenging enough that it can still be used legitimately in fiction.
Twenty Questions, a game where, counting the first question (usually if it's animal, vegetable, or mineral), the players can ask no more than twenty questions to guess what the active player is thinking of, and all questions after the first must be the "yes or no" kind. Usually parodied now instead of played straight. Computers can play it quite well, e.g. 20Q and Akinator.
I Spy, a guessing game similar to but even more basic than Twenty Questions. The active player thinks of something within their line of sight and tells everyone else its color or first letter. They try to guess what it is. Only ever played by really bored characters.
Musical Chairs, is usually just played in children's parties now. Someone sets up enough chairs for all but one of the players to sit on. They walk in a circle while some music is played for a short time. As soon as it stops, everyone tries to sit in a chair, often resulting in a Big Ball of Violence. The one who can't is out, and one chair is removed for the next round, until one chair is left, and the one sitting is the winner.
Charades, nowadays the lowest of these games in fiction. Unless it takes place in the past, it rarely is portrayed for any reason other than to show what losers the players are. It is played by acting out the words the active player is thinking, puns and homophones allowed. The only other clue was to hold up a finger for each word in the answer, and fingers for which word is being played. Such improvised Hand Signals are sometimes used by a character to attempt to convey information which for whatever reason, such as being mute, they cannot simply say aloud.
- Pictionary is Charades with drawings, where one partner must draw the clue instead of acting it out. This variant is more common in animation for obvious reasons. Sometimes the drawings aren't seen by the viewer.
Blind Man's Bluff, is usually seen in portrayals of older times. One player is blindfolded, while the others hide. The blind man has to find the other players. This game is sometimes depicted as a flirtatious man looking for giggling young women in a parlor.
- Marco Polo is a variation on Blind Man's Bluff, with three differences: a) It usually takes place in water, such as a pool. b) The hunter doesn't wear a blindfold, but rather just keep their eyes shut. c) Most importantly, the hunter can call out "Marco!" as often as they like, and if the hunted ones hear it then they must respond with "Polo!". It's pretty much a miniature version of submarine warfare, sonar and all.
Truth or Dare is stereotypically most common at a slumber party, but can take place in other situations as well. The very point of this game is to elicit personal revelations if someone picks "Truth," or wacky hijinks if someone picks "Dare"; therefore, just by playing the game normally, it's quite likely that the events of the game will generate results interesting enough to be the plot of a story. Fanfic writers know this very well, and Truth or Dare fics are practically a genre.
I Never is a similar game to Truth or Dare. Usually played more as a drinking game, although other forfeits are common. The premise is for one person to say something which (hopefully truthfully...or not, as the case may be) they have never done, and all the other players have to commit the forfeit if they have done that thing.
Spin the Bottle and Two Minutes in the Closet (or whatever variation) are the classic young-coed-teen-party games. In the first one, sit in a circle, take turns spinning a bottle and kiss the first member of the opposite sex it points to. In the second, pull names/number out of a hat to form couples and go into the closet for two minutes and... amuse yourselves in some fashion. This is often a way to trap/nudge a character into his/her First Kiss, to set up/exacerbate romantic jealousies or to contrast different levels of sexual activity among a bunch of kids of the same age. There will be much awkwardness, blushing and wiping of sweaty palms.
Mafia divides the players into two teams. One team is initially much smaller than the other, but the composition of the teams is unknown to the members of the larger team. The game alternates between turns during which the larger team keep their eyes shut, allowing the smaller team to communicate in secrecy, and turns during which all players claim they belong to the larger team. The elimination of a player is debated every turn. Paper sheets or cards are often used to create the teams at the beginning and to "unmask" any player who was just eliminated. A referee is normally required. Furthermore, a single player of the larger team has a hidden turn of his own, during which he learns the true allegiance of another player. Additional roles and teams can be introduced, potentially leading to at least one Double Reverse Quadruple Agent. In fictional works, Ten Little Murder Victims will sometimes play this kind of game right before it becomes the plot.
- The Werewolves of Miller's Hollow is one of many commercial versions, with a dedicated deck of cards.

Examples of Parlor Games include:

Literature

Jane Eyre features what is most possibly the most elaborate game of Charades ever. They make sets.
Scrooge and the Ghost of Christmas Present observe Twenty Questions being played at Fred's Christmas party in Dickens' A Christmas Carol. The "animal" in question turns out to be "Uncle Scrooge".
- In the movie version with Patrick Stewart playing Scrooge, the game is Blind Man's Bluff.
- In the George C. Scott-as-Scrooge adaptation, the party is playing "Similies". Fred would say the first part of a common expression, such as "Quiet as..." or "Tight as...", which the player would then have to fill in (in these examples, 'a mouse' and 'a drum', respectively).
In Thomas Pynchon's The Crying of Lot 49, Oedipa and Metzger play a game they refer to as "Strip Botticelli" while drunk, but they don't actually get that far and Metzger ends up taking his clothes off anyway.

Live Action Television

Seen It a Million Times in Sit Coms.
Wings used Simon Says as a plot device, and an Actually Pretty Funny moment.
At one point in Buffy the Vampire Slayer, Spike and Harmony play Twenty Questions. It's a breadbox.
Kate and Sawyer play I Never in a Season 1 episode of Lost
There was an episode of Sabrina the Teenage Witch wherein in order to undo the wacky spell-gone-awry of the week, Sabrina had to get someone to say a certain phrase—but of course, Sabrina wasn't able to actually say it herself, because that would make things too easy. So instead, she initiated a game of charades and tried to get the person to say the phrase that way.
In the Doctor Who episode "The Unicorn and the Wasp," the Doctor has been poisoned with cyanide, but can cleanse his system with only a few ingredients. Unfortunately, his mouth is full when he needs to tell Donna what he needs and a hilarious game of charades ensues:

Donna: I can't understand you... How many words? One! One word! Shake...milk-shake...milk?! No, not milk. Shake, shake, shake?! Cocktail shaker! What do you want, a Harvey Wallbanger?
The Doctor: HARVEY WALLBANGER?!
Donna: Well, I don't know!
The Doctor: How is "Harvey Wallbanger" one word?!

Two games of Charades were played in a courtroom sketch in a Monty Python episode. Go here and search for "We have, m'lud."

We find the defendant Not Guil-cup.

Michael on The Office plays Charades in "Dinner Party," but he can't resist cheating. "Rhymes with Parnold Schporzenegger."
The Three's Company episode "Jack Bares All (aka Oh Nurse), Part 2" has Mr. Furley introducing a game where each player makes up a question one word at a time. They don't like it, so they play a game in which everybody has to imitate another partygoer. Hilarity Ensues.

Film

The villain of Die Hard with a Vengeance used Simon Says in his games.
Twenty Questions was parodied in Bill and Teds Bogus Journey. It was a tank, and Ted guessed it in two questions.
- "Are you a mineral?" "Yeah." "Are you a tank?" "Yeah!"
Nephew Fred and his party play Twenty Questions in the 1951 version of Scrooge.
In Young Frankenstein, Frankenstein is strangled by the monster, and tries use charades to tell Igor and Inga to administer a "sedative". Their guesses include "said-a-give" and "said a dirty word".

Frankenstein: SEDA-GIVE?!

Demolition Man: Big Bad Simon Phoenix references Simon Says as a sort of Catch Phrase:

Phoenix: Simon says bleed!
Phoenix: Simon says die!

Mozart and his wife play Musical Chairs at a party in the film version of Amadeus.

Other

And when Adam West played Twenty Questions with himself on Family Guy. "Am I Bo Bice? Yes, I am."
FoxTrot had a comic where Roger and Andy played pictionary. Andy drew what was very obviously a boat, but Roger struggled to figure out what it was, suggesting such things as "a Christmas tree in a cereal bowl". When she wrote "boat" at the bottom of the page, he thought it was some kind of "pictionary shorthand".
In this strip of Wapsi Square, Katherine wants information from Monica who is barely capable of speech before her morning coffee. This results in a game of charades.
In the musical Evita, Juan Peron and other generals play a game of Musical Chairs to the music "The Art of the Possible," symbolizing Juan's rise to power in the chaos of post-revolutionary Argentina.
Given a very spooky slant in Dark Fall: Lost Souls. The fact you're playing them with a Creepy Child ghost doesn't help...