Whether the following is an "English game" is up to you, but I remembered something from my childhood.
Back in primary school (in India), we kids sometimes used to play in our free time a game called "Name, Place, Animal, Thing".
This was a game we played on our own, i.e. it was part of the children's subculture, not something taught by teachers.
The game was for at least 2 players (typically 4 to 8: you could make groups if you decide to do this in your class), and went something like this.
Each player makes four five columns on a page, labelled "Name", "Place", "Animal", "Thing", and "Score".
A random letter of the alphabet is picked.
(E.g., one player recites in the alphabet in her head, and another says "Stop" at some point; the first player announces the letter she was on.)
Each player then writes down a "name" (proper noun), a "place" (geographical), an "animal" (may include birds, etc.), and a "thing" that starts with that letter .
You can't use words that have already been used in the game (i.e., when the same letter came up earlier).
After everyone is done (more fun if there's a time limit; typically the player who finishes first counts slowly to ten), each player is asked to read out his "name".
Repeat with "place", "animal" and "thing".
Score :
If a player has picked a name that no one else has, she gets 10 points.
If two players have picked the same name they get only 5 points each.
(Or 10/(number of players) if you wish.)
Repeat for "place", "animal", and "thing".
So each player can get up to 40 points for that turn.
Obviously the strategy — beyond knowing at least some word with that letter — is to pick unusual enough words that no one else is likely to pick.
More details on the game here.
Unrelated aside: Apparently some American kids are taught to categorise all nouns as "person", "place" or "thing".
Here's an article: The Noun Game –
A Simple Grammar Lesson Leads to a Clash of Civilizations .
english.stackexchange.com/users/77/ShreevatsaR
In a Pickle .
For up to six players, though I don't see any reason why it couldn't be played with more people.
Can a giraffe fit in a phone booth?
Does a sofa fit in a shopping cart?
It all depends on how you size it up in this game of creative thinking and outrageous scenarios.
Try to win a set of cards by fitting smaller things into bigger things — there's some juice in a pickle, in a supermarket, in a parking lot.
Play the fourth word card to claim the set, unless one of your opponents can trump with a larger word.
The player with the most sets at the end is the big winner!
Includes 300 cards and instructions.
I can tell from personal experience that the game is quite fun, for native and non-native speakers alike.
It encourages thinking outside the box, re-interpreting words and defending your interpretation.
Someone puts a president in a jam, but then someone else puts that jam in a glass.
Or someone plays the card with the word "universe", thinking that no one can beat that, but then the next person plays the card with the word "dictionary" and wins the set (after all, "universe" is a word in a dictionary).
english.stackexchange.com/users/300/RegDwigнt
Mad Libs are fun and could be instructive regarding parts of speech.
You could add complexity using descriptions like "past participle" or "first person plural verb in future perfect tense".
english.stackexchange.com/users/2717/Mitch Schwartz
The following page has a number of games: Games for English language learning and teaching
english.stackexchange.com/users/127/Mehper C. Palavuzlar
I always recommend Taboo!
(doesn't have to be the official version) as a language learning tool.
The game is about communicating a word without using certain related words, so the skills it develops in thinking of ways to communicate it are useful when you're in a conversation and your vocabulary comes up short of something you want to describe.
english.stackexchange.com/users/3542/Peter Taylor
Dictionary is a fun game -- you can buy it, or just play it with paper, pens, and a dictionary.
You need at least 3 or 4 people, up to as many as you wish.
One person picks a word out of the dictionary that no-one else knows.
Each person writes down the word on their paper.
The person who has the dictionary writes down the meaning, everyone else makes up a definition.
All had their papers to the person who picked the word.
That person shuffles the papers and reads each definition out loud.
Each person chooses what they think is the correct definition.
When all have made their choice, the person who picked the word gives the correct definition.
He or she gets 1 point for everyone who did not select the correct definition.
Anyone else who had a definition that was selected as correct gets a point for each person who chose it.
And everyone who selected the correct definition gets a point.
Then the dictionary is passed to the next person, and another round is started.
The game is over when all have had a chance to pick a word (because you tend to get more points when you pick the word).
The person with the most points wins.
This encourages trying to make up definitions that sound real, or making definitions that are funny (no points, but lots of laughs).
english.stackexchange.com/users/728/thursdaysgeek
I'll be hosting an English activity this week. Could you share some interesting English games which impressed you the most?
Simple yet interesting English game for class?
english.stackexchange.com
<word-games><teaching-aid>
english.stackexchange.com/questions/7374
english.stackexchange.com/users/2796/ZZcat
There are a wide range of English games that you could use in a class. Some examples include "Name, Place, Animal, Thing", "In A Pickle", "MadLibs" or "Taboo". There are several games that can be used to teach English. Some sample games are "Name, Place, Animal, or Thing," or "In a Pickle", Pictionary, Taboo. They test various different elements of the English language. Some can be bought or simply devised using pen and paper.
There are a wide range of English games that you could use in a class. Some examples include "Name, Place, Animal, Thing", "In A Pickle", "MadLibs" or "Taboo". Various games were suggested, including "Name, Place, Animal, Thing", "In a Pickle", "Taboo!" and "Dictionary".
Suggestions include Mad Libs, Taboo and Dictionary. There are a wide range of English games that you might like to choose from, but some suggestions include "Mad Libs", "Taboo" or "Dictionary".
There are several games that can be used to teach English. One was called "Name, Place, Animal or Thing".  It is a type of guessing game where the players earn points. Another game is called "In a Pickle". This is a card game Mad Libs is another game that tests knowledge of verbs. Taboo is another game. It involves communcating the word without using related words. Dictionary is another fun game, which can be bought or devised oneself.
Name, Place, Animal, Thing. In a Pickle . I always recommend Taboo! S0: Dictionary is a fun game -- you can buy it, or just play it with paper, pens, and a dictionary.  
There are a wide range of English games that you might like to choose from, but some suggestions include "Mad Libs", "Taboo" or "Dictionary".