depends of game type. you can't generalize really except for aspect of challenge.
if we are talking about pvp games, these games must have competitive gameplay on a good level. Example of a good game is Counter Strike or Quake, when example of bad games are battlefield and call of duty.
In mmo games, balance of weapons, classes for a fair pvp. In a good pvp game players as acommunity create their own content, artificial content which keeps them engaged for a very long time. Like a 'Fury' server of Age of Conan mmo game. When basically 3 guilds were carrying the game for two years straight.
if we are talking about pve games, we face another, more complicated situation. pve game must have plot, challenge, goal, game design, mechanics, atmosphere, characters. from these aspects fun is being born. because these aspects form a content of the game.
be it a single player game, coop or mmo game. most, if not all, of these aspects should be on a decent level.
Unfortunately for the last decade we may observe less and less products, which can be classified as a good games.
p.s. hate to bring this up. quality of the end product is also important. Because in these nights, a lot of developers and publishers deliver a products which are not ready to be released.
anyway, in short: game must challenge me.