Hmm, interesting...what different kinds of "games" is this word used for? Would "Monopoly" (the board game) be ένα παιχνίδι? What about a football match on television? Or when children play "hide and seek"?
Monopoly is considered ένα παιχνίδι but to be more specific you could call it ένα επιτραπέζιο παιχνίδι (a board game) επί = on,(το) τραπέζι = table. You can say to your friends "Ας δούμε το παιχνίδι" = "Let's watch the game" or "Ας δούμε τον αγώνα" (αγώνας translates to fight/match). And finally, yes hide and seek is considered ένα παιχνίδι as well. It's also used between male (mostly) friends as a way to say "I want to get to a relationship with her", "Θέλω να κάνω παιχνίδι μαζί της". Θέλω = want, κάνω = do, μαζί = with, της = her Electronic games are also considered παιχνίδια Hope this helped
Monopoly would be referred as an επιτραπέζιο which is the exact translation of a board game. The other two examples about a match on television or hide and seek are translated as παιχνίδι . Hope it helped!