This first definition might be behind the BGG category of 'political' games, though I'm not entirely sure - I guess you could say Coup kind of has a political theme (its original incarnation certainly had one), Twilight Imperium has that a phase explicitly termed 'political' and Warrior Knights has that piece with the parliament where you vote on laws (though so does Lancaster, which isn't categorized as political - not to be confused with Wars of the Roses: Lancaster vs. York that is categorized as political, though its parliament phase has no voting and is not very... well, not very political). So these games have themes with political flavor. But Diplomacy's theme is basically war, not politics - it is won by dominating the map not by passing laws. But even if it isn't, there's something missing in saying that Diplomacy is a political game because one of the thing it simulates is the diplomatic relationship between countries at war. Diplomacy is a political game because of what it encourages us to do.
Which leads us to the second definition: a political game is one whose core mechanics create or encourage a certain kind of interaction, which I will for now term 'political'. This is a bit vague but bear with me for a bit here while we explore this. This the kind of games that I find even more interesting, and will be my focus here. Now, it's important to note that these categories of games are not mutually exclusive: Tammany Hall, for example, is certainly both. But what kind of interactions we think of as 'political'? What's special and great about this kind of games? What's horrendously awful about them?
You might be reminded here of my discussion of political analysis of games. I said there that what we focus on when we analyze games politically is the way people relate to each other, and specifically the powers they have over each other. And that's the first clue as to what is a political game - it is a game that encourages, or requires, people to do stuff to each other. More specifically it's a game where players stand in relations of power towards - they have specific things they can do to each other. Often, these powers are encapsulated in roles that are basically titles for specific game states that bestow powers on one or more players.
Now that's still a pretty general statement. And in some ways, it's true about every game. In Splendor, players buy cards from the main rows that are offered to everybody. It is not a bad idea to try and snatch away cards that other players want, not to mention the race for the nobles. But Splendor is not, or so I think, a political game. Don't get me wrong - I like Splendor. And though it's light on player interaction (on in general, a light game) - I don't consider it 'mutliplayer solitaire'. But the line has to be drawn somewhere. In Splendor you only do things to other players by buying cards or achieving your own goals. This ability is equally distributed among players and among all strategies. The only thing you can do to another player is ruin their plan by taking what they need and using it in the building of your own project. And that isn't much, essential as it may be for the superb Splendor strategist.
It seems like this is not a dichotomous criterion. There is definitely a continuum - some games are more political than others, and the more political games allow (or encourage) you to do stuff to other players. Let us go back to our Diplomacy example. In Diplomacy, the main thing you can do to other players is attack them (you can also support them or ferry their armies across water). That's not very political, in my opinion. But you are also encouraged to make deals with them because the equality of units on the board and the geographic location of some powers (Ahem, Austria), means that you absolutely cannot go for it alone. At the same time, the game provides you with a great opportunity to renege on these deals because orders are submitted secretly. This mixture of conditions has people doing all sorts of things to each other aside from attacking - promise, threaten, cajole, beg, reproach and just plain old manipulate to get what they want. Furthermore, the fact England needs the help of France but cannot know for certain the French are telling the truth means that the French has power over England. Depending on the board position, it might be quite a bit of power - as players often depend on other players for their survival. And that's a lot more interesting than just killing each other.
It's easy to think that this is all political gaming is about - people needing things from each other and therefore making promises that they don't intend to keep for the ever-sought-after-but-never-really-achieved perfect backstab. But good political games go way past that and play around with the interesting things that people can do to each other and the powers that they have over each other. In Diplomacy there are no special roles explicitly assigned to players, but the dynamics of the game often make it the case that one player becomes your ally, and therefore stands in a special relationship to you. The fact victory can be shared also encourages that kind of long term partnership. But the same idea could be played around with in interesting ways. In Dune (aka Rex) there is a Nexus phase that comes up randomly every few turns which is the only time alliances can be formed. Alliances are formal in Dune and allies win and lose together; once you're in an alliance, you cannot break it until the next Nexus, which means you really care about your allies' condition. A good Dune player will jump into and out of an alliance in opportune moments, but that's a risky and complicate trick to execute, and requires quite a bit of skill.
Dune is still an area control game where combat is a main component. Consider Tammany Hall again - it's a game that has much more of a Euro feel, a la El Grande. It's an area majority (rather than area control) with much fewer opportunities for direct confrontation (though you can remove opponents' pawn with 'slander', it is costly and comes at limited supply). The theme is political, but it is to a great extent a game of managing resources - how do I win most areas with the least amount of pawns in each?
But then the game makes the brilliant, and political, twist. The theme may be political but that doesn't make the game any more political (El Grande technically has this 'war' theme with your cubes being 'knights' but it doesn't feel anything like war). Two mechanics draw out the political aspect. First is the simultaneous blind bidding during election time. Following the footsteps of Diplomacy, discussed above, it makes players depend on each other and encourages horse trading in a distrustful environment. Second, it has the cool mechanic of roles assigned by the mayor. When you win election, you get to be mayor which gives you some valuable victory points. But then you, the player, assigns other players to offices that give them special abilities. And these are quite important for success in the game. Now the designer could have said that whoever comes in second gets to be the Deputy Mayor, the third is Council President and so forth. This would have made for a more strategic, yet less political game - it would have allowed the players to strategically aim for a certain role by coming in third or force. But the fact it gives one player such interesting power over the others makes for a game that's much more politically interesting.
Political games are not the only games that I like but they hold a special place in my heart. I love strategy and the competition that more deeply strategic games allow. But its the Coups and Sheriff of Nottinghams that really make for interesting interactions between people rather than between strategies employed by people. I usually don't like playing political games with people I don't already know and trust, because they can really impact relationship and are much more fun and interesting when they're used to explore relationships. If you're interested in board games because of the interactions between people, political games are for you.