What makes a good game? Why are some games fun and interesting, while others are just meh? Here are some criteria, in no particular order, that I use when I try a new board game or card game:
1. Easy to Learn
The game has to wow me right off the bat. Now, some games may have a neat gimmick (like Killer Bunnies) or take place in an interesting setting (like Warhammer 40,000 or Betrayal at House on the Hill), and for such "fluffy" games, I'm willing to be patient and learn a complex rule system.
For the most part, though, I enjoy a game that I can pick up and learn easily. It has the advantage of not losing my attention, which can be a bit short, span-wise. Also, I can turn around and teach the game to others.
2. Depth of Strategy
Easy to learn, difficult to master. Those are traits I really enjoy in a game. Card games like Magic: The Gathering or Dominion, for example, have a lot of subtle interactions between the cards. Similarly, the base mechanics of Carcassonne are easy to grasp, yet the game provides many strategic options and a plethora of decisions to make.
A writer and game designer whom I admire greatly once wrote an article about the difference between choices and decisions in a game. It took him an entire article to fully explain his thesis, but the gist of it was that some games provide you with many choices, or options, and you simply have to choose the best option. Other games, however, create moments wherein your options suddenly become limited, and you must make a decision about which route to take.
I believe that a good game backs players into corners and forces them to make strategic decisions. "Do I sacrifice my bishop to put myself on offense, or do I just move my king out of immediate danger, but stay in a defensive mode?" That's a game-changing decision, and it adds tension, which I enjoy. "What do I kill with this Lightning Bolt?" on the other hand, is a choice, which may create a decision for my opponent, but is usually just simple mathematics for me.
Some games have a very heavy story element to them, which is cool. It can be harmful, though. It's hard to justify spending sixty dollars on a game that's only good for a couple plays. Murder mystery games, for example, suffer from this problem. You play through them once, and from then on you know whodunnit. Even if you can get a completely fresh group to play with you, you already know the story. The Horus Heresy game, I am told, has the same problem. While it may be fun to play out the final battle to defend the Emperor of Mankind against the corrupt heretic forces, after six or seven games, it gets stale.
Single-player video games can suffer from this same problem, although in their defense, they often take a few dozen hours to finish. Fifty dollars for a game that takes twenty five hours to beat equals two dollars per hour of entertainment, which makes it four or five times better than hitting up a movie theater, as an investment. Board games that lack replayability, however, can sometimes only boast five or six hours of fun before becoming uninteresting. At that rate, you'd be better off taking your sweetheart to a water park.
4. Rewards Skillful Play
I enjoy feeling rewarded for my efforts when I'm trying hard and playing well. There are some games--Killer Bunnies comes to mind--that, when boiled down, are almost entirely random. Roulette is the same way. You're just hoping to get lucky. I don't care for that. I like feeling that my efforts are paying off.
With games like Carcassonne or Magic: The Gathering, on the other hand, you can actually watch as you gain points on the score chart or see your opponent's life total diminish. It's fabulous.
As much as I like being able to play skillfully and be rewarded, I also like a bit of randomness in my games. Chess is all well and good, but the really good players know when and how it's going to end long before less-experienced players such as myself. A bit of randomization, like rolling dice or shuffling cards, helps balance the game enough that newer or less-skilled players can still have a shot at winning. It adds a bit of danger and excitement to the game.
6. Non-Aggressive Strategies
This isn't as important to me as some of the other points I've mentioned, but I am not a mean person at heart. Sometimes, I have trouble being cutthroat enough to win games. I like games like Dominion, Carcassonne, or even (if you play your cards right) Magic: The Gathering, that provide strategies wherein you don't actually have to interact with the other players. That way, I can avoid upsetting anybody. That's probably why I love combo decks in Magic: The Gathering so much. I can just do my own thing, ignore my opponent, and try to just win before my opponent can kill me. Although that can upset some opponents, too, so I guess I can't always avoid making people mad. Sigh.
This is hard to quantify. Different people enjoy different things.
Here's my beef with games like FarmVille, Mafia Wars, or World of Warcraft. You do the exact same thing over, and over, and over again. Now, that would be fine if what you were doing was enjoyable, but it's just... not.
I sure do loves me some Dominion, though.