For example, Monopoly is a game, and I own a copy of it. When I am playing it, I am playing a what of Monopoly? I'm looking for a word that highlights that the game is in progress and differentiates it from the concept of the game.
From the Free Dictionary:
2. a. A competitive activity or sport in which players contend with each other according to a set of rules: the game of basketball; the game of gin rummy.
b. A single instance of such an activity: We lost the first game.
To indicate in progress just use present continuous tense:
"We can't come outside right now, we're playing a game of Monopoly."
A round is one word that is used.
For those unfamiliar with the use of round, see:
Oxford English Dictionary
VII. A period or bout of play at a game or sport, and related senses.
34. a. summary: a game of cards = a round of cards
b. summary: refers to a round (=game) of golf
c. gen. A bout of play at a game, sport, or contest; (in early use) spec. a bout of fisticuffs. Also in extended use.
1962 E. Albee Who's Afraid of Virginia Woolf? (1965) ii. 85 We'll play a round of Get the Guests. How about that? How about a little game of Get the Guests?
1994 Rotarian Sept. 4/1 Planning a visit to a sister club in the Philippines, and playing a round of mah jong, were far from regular, bland tourist staples.
2009 M. Herczog Frommer's New Orleans 2009 xi. 277 Enjoy some shepherd's pie, wash it down with a sample of the nice selection of beers on tap, play a round of pool or darts.
THUS: a round of Monopoly
You may not say it like that, but me and my kind do.
Hmm... "round" has other meanings within the context of a game. I agree that the term "game" can be used to refer to one instance of play, however if you need to differentiate between "game" (the collection of rules and/or objects) and "game" (the instance of play), then I think the most appropriate term is "session".
As has already been mentioned, a game of Monopoly is the term we often used. You could also say "We were playing monopoly." A game consists of several turns.
The article used and the context is what differentiates. "A game of Monopoly" refers to a round/match of "the game Monopoly." However, if you are referring to a specific instance of the game, you would go back to using "the" as in "Mark won the game of Monopoly that we played yesterday when I landed on his Illinois that had a hotel on it."
I'd be comfortable calling this a run of a game. "Bill won the last run of Monopoly" sounds perfectly natural to me. This usage comes from the form of the word meaning "a journey, especially one taken repeatedly", making it particularly effective for differentiating between multiple distinct game playing sessions. This might also qualify playthrough for the same reasons.
These terms only work for games with a well-defined beginning, middle, and end. However, some video games for example are "sandbox" style, open-ended experiences with no specific end. Referring to a session of play for such games as a "run" or "playthrough" wouldn't make sense, as the point of these games is to have a unique journey each time you play them, not to repeat a journey.
One can sometimes see "a play of the game". This may be rare compared to the other answers we have seen here.
A game of chance offers the following odds and payoffs. Each play of the game costs $200, so the net profit per play is the payoff less $200.