I'm wondering if LittleBobbyTables' definition here, might be a good, simple, definition of what this site covers?
What potential grey lines, or exceptions might there be? Are they serious enough to make it unworkable?
There are a few axes on which we need to figure out where to put the dividing line between on and off topic. For instance, the rule you mention, can it be played on a table, is a good one, but not sufficient, as you can sit down and play an RPG on a table, but I think that's off topic for this site.
Here are a few areas where we need to figure out exactly where to draw the line. To the left are what I would consider solidly on topic, to the right probably off topic, and we need to figure out where the line is in the middle.
For the first category, I think that solitaire games are on topic, while puzzles (such as Sudoku or crosswords) are off topic. But how to distinguish solitaire games from puzzles? I believe the criterion would be that solitaire games essentially produce an endless or large variety of different puzzles through randomization (most solitaire card games involve just shuffling a deck of cards, solitaire or coop games might involve rolling dice or other forms of randomization), while puzzles are static and new puzzles in a particular category (such as Sudoku or crosswords) must be created by another person or computer. I would, however, say that questions about puzzles that are specifically related to a board or card game, and commonly used in training for playing board and card games, would be OK; for instance, "I'm working on this chess problem, and the book lists this solution, but I just can't figure out why this other line of play doesn't work."
For the second category, I think the distinguishing criterion is that there are objective rules and win conditions in a board game. The win condition or outcome of a situation might be based on subjective votes or other actions from other players, but once people have made their decisions, there is a set of rules for determining the winner or outcome of a situation. In a roleplaying game, a GM is needed for deciding which rules to apply, running NPCs, etc, and so there is no objective set of rules for resolving situations or the outcome of the game.
In the third category, I believe that dexterity games can be on topic (such as Jenga or Pitch Car), but I think that the rule you mention, that you can fit them on a table, is a good one. Thus, crokinole would be OK, but foosball or pool would not.
In the last category, I think that the rule that you can play it by hand, with no computer involved, should apply. You may use electronic equipment to aid you (an electronic dice roller in place of physical dice, or using a spreadsheet to keep score instead of doing it by hand), and questions about how to do so may be on topic, as long as it's all stuff that you could do by hand. You can't play a game of Quake by hand, so questions about Quake are off topic; but you can play Ticket to Ride by hand, so asking about places to play ticket to ride online are OK.
Or to use Pat's example, Civilization and Advanced Civ are clearly on topic, Sid Meier's Civilization is not, as it can't be implemented as a boardgame. The computer game port of Advanced Civ would be on topic, as it's a game that could be played on a board, though for technical questions about it there may be better forums (or may not; people on a boardgame question and answer site may have more familiarity with the online forms of the games than people on a computer game or computer software question and answer site, and a good measure of whether something on topic is whether this is the best place to find people who can answer a question about it), and Sid Meier's Civilization: The Boardgame is on topic, as it's obviously playable by hand on a board instead of on the computer (and if anyone ever implements Civilization: The Boardgame: The Computer Game, asking about that would be on topic too, as it's still playable on a board).
Do those seem like a reasonable set of guidelines? In summary, for a game to be on topic, it must:
For a question to be on topic, it must relate to a game that is on topic, but the particular instance that the question is referring to may loosen those rules. Questions about playing games with giant pieces will make a game not playable around a table, but if it's a game that could be or is normally played around a table, it's fine. Questions about Go problems or Chess problems are fine, even if they are static puzzles, since they are related to a dynamic game. Questions about computer implementations of board games are fine.
I think that gives us a pretty good definition of what's on and off topic; there will always be some grey zone, but it will solve most of the problems.
To expand on my comment above, we would want to consider these four games. Which of them are on topic, and why?
One rule I would definitely have is that anything with a computer-programmed AI opponent is banned.
Computer games, where they are really electronic versions of boardgames - and where there are actual copies of the game made from atoms not bits - are at least arguable. Certainly, the majority of questions relating to Diplomacy are the same whether it's FTF, PBM, PBEM, DiPjudge or phpDiplomacy, and I think very few would object to questions relating to black-press, public-press or gunboat (all variants that are only possible in non-FTF formats). Where it's a commercial implementation of a straight conversion (e.g. Matrix Games' Empires in Arms) then it's more arguable, but again if the question is about the game and not the computing aspects, then I think it's on-topic.
The only AIs that we should be discussing, though, are the scripted pen-and-paper AIs, like the solitaire game of Republic of Rome. I guess a computer implementation of a pen-and-paper AI might be acceptable.