An ambiguous overloaded phrase with several common meanings, its definition is the subject of many endless forum threads. There is the similarly ambiguous variant term "abstract strategy game".
In some sense, it boils down to whether one considers the phrase "abstract game" to be literally a sum of its parts (i.e. a game which is abstract), or if one considers the phrase "abstract game" to have picked up a specific meaning through use (e.g. how a "wargame" is typically considered to be more than simply a game about war, so that many people would not consider the card game Nuclear War or Battleship to be wargames).
It also depends on what one thinks "abstract" refers to - the mechanics, the theme, something else?
"Abstract game" often refers to games of perfect information with alternating turns and no randomness; usually for 2 players. E.g. Chess or Go.
It is also often used to refer to a game without a theme (regardless of the game mechanics). E.g. Backgammon or Tantrix.
It is also often used to refer to games that are "mostly" perfect information and "mostly" lacking in theme (leading to even more confusion and disagreement about whether a given game is an "abstract game" or not).
E.g. many people disagree whether the following are "abstract games": Stephenson's Rocket and König von Siam and many 18xx games, since they have perfect information and no chance, but a clear historical theme.
And many people disagree whether the following are "abstract games": SET and Poker, since they have no theme, but randomness, hidden information, simultaneous turns, realtime turns, etc.
There is also disagreement about whether some games even have a theme, e.g. whether Chess is themeless or has a theme of a medieval battle, or whether Cathedral is themeless or has a theme of building a medieval city.
2-player games with perfect information, no chance, and alternating turns are sometimes more clearly called "perfect information", "pure strategy", "non-random", or "Combinatorial games" (the mathematical term for such games). Note that such games can have a theme or not.
Games with no or little theme are sometimes more clearly called "themeless" or "weakly themed" instead of "abstract". Note that such games can have luck, hidden information, simultaneous turns, or not.