All games, when it comes right down to it, boil down to one or more of these four simple elements. Strategy, Randomness, Hidden Information, and Social Factor.

Strategy is involved in a game when you can study the board, mentally work out the best move available to you at the time, what your opponent is likely to do next turn, and ideally you can plot out elaborate tactics over the next several turns. There are many games, particularly in the list of games which have existed for centuries, which contain no elements other than strategy. Tic-Tac-Toe, Chess, Checkers, Go, and Mancala all come to mind. The key to injecting strategy into a game is to ensure there are enough options at any given time that the best possible choice isn't completely obvious. Tic-Tac-Toe is an excellent example of how pure strategy can be bad. Any game of pure strategy can be "solved." Every possible sequence of moves can be examined, yielding a single specific strategy which a player can follow and never lose. For most games, such a strategy has not been found yet, but it is out there. Meanwhile, whichever player has the most experience with the game and raw processing power is almost definitely going to win if strategy is the only factor.

Randomness is involved in any game where you roll dice or draw cards. Making a game where randomness is the only factor can never work out, but many try anyway. Board games aimed at the very young often involve nothing at all but rolling dice, and several simple card games, such as War, or Five Card Stud (without gambling) also qualify. In all of these cases, the players have absolutely no effect on the game whatsoever. If we're playing Five Card Stud, and you flip my hand over for me as I am out of the room, the same cards will be revealed as if I did it myself. There is no real game if randomness is the only element.

Hidden Information is involved in any game where you know something your opponent doesn't, and attempting to deduce this information, or keeping it a secret, has some major impact on the game. Paper Rock Scissors, Odds or Evens, and Pick a Hand are the only games which come to mind involving nothing but hidden information. This can either result in something completely random (which we have established is bad), or become a complex game of psychological warfare, where each player draws upon their knowledge of how the others think, and second guess each other, much like the poison scene in The Princess Bride.

Social Factor is involved in any game where players can identify who is winning at the moment, and gang up on that person. I can't think of a single example of a game which only involves social factor, so I will have to create one just for these purposes. Lynch Mob. Social factor has an unfortunate habit of overriding any other elements a game may involve, and rewards you for sitting quitely in the corner while punishing any attempt to up your score.

There may be other things which one could potentially consider gameplay elements were they seeking to make a more complete list, skill for instance, which is the sole factor in a game like Darts. Any such things however tend not to mesh with those elements listed above. At least, not in any way which doesn't feel so forced as to hurt the appeal of the result.

It is my personal opinion that for a game to be truly enjoyable, it must involve at least two of these elements, if not more, to some degree or other. Some combinations of course work better than others. Strategy and random factors for instance work very well, as seen in more or less any tabletop strategy game. Strategy and hidden information work nicely too, just look at Stratego, or a computer strategy game featuring "Fog of War." All three work extremely well together too, but random factors and hidden information without the others work terribly together. Social factor meanwhile can make things more interesting in small doses, but frequently ovewhelms everything else, causing a game to effectively become a one or two turn game of Kill Dr. Lucky as soon as someone pulls ahead, negating everything which happened up until that point. This is a serious problem with the industry as of the time of this writing in my opinion, not so much because these games aren't fun (I deeply enjoy Kill Dr. Lucky and the majority of these games which degenerate into it), but because it creates a homogenous crop of games, and in comes the bordom.

Personally, I like to shoot for a mix of the first three elements in my games, primarily focused on strategy as the deciding factor of victory. With this mix, I can play against the same person dozens of times, and never be sure exactly how the game will go, or who will win, but I will know that if I'm doing well, it's because I have a good grip on the mechanics. In other words, lasting fun.