Often, one plays a computer game through once and then is done with it. "Replay-ability" is something that is a great plus with a game, but not every successful one has it. But with board games, being able to play again and again is a basic necessity for its success. And some games are still loved (e.g., Monopoly and Risk) generations after they were first invented. The ultimate in replayability. Why is it that some games continue to hold lasting appeal even when society, culture, and the people themselves have changed so much since they were invented?
I agree with Lo'oris' argument that most older well known games are still widespread mostly because they already are everywhere. How many people go out and buy chess boards compared to the number of people who simply have them around from previous generations?
I think there are some additional aspects:
In my experience games need a good balance between complexity and depth in order to be popular. Basically the ratio of options a player has divided by the amount of rules he has to learn. A lot of games that are popular in our family are relatively easy to learn but offer a lot of possibilities in comparison. This can be illustrated especially well with games that are in very similar but somehow one of them just feels better. Compare Pachisi and Dog/Tock. The have the exact same basic concept — move your pieces around the board one time and get into the safe haven. But in all the groups I've played in Dog/Tock has completely displaced all other Pachisi variants because with a few additional rules it has become a much more interesting game.
Most modern boardgames aim to produce a specific atmosphere or experience that is perhaps unusual for the players — a feeling they would otherwise not have. With so many games out there a lot of times games can feel similar to each other. There are countless games where you throw dice, move along a path and do something according to the field you landed on — nothing special. And then there are games that somehow manage to capture something special. For example Pandemic expertly captures the rush of working together against an overwhelming threat, Bohnanza produces an intense trading atmosphere, and games like Werewolves or Battlestar Galactica allow people to play out their mischievous and paranoid traits.
Whenever a game is released that succeeds with a good combination of these two (and maybe other) aspects then it is likely to become quite popular. Of course others will try to copy and improve that mixture. Some will fail but others will eventually succeed. At that point a battle will take place between the improved qualities of the newer game and the established following of the older one.
Monopoly and Risk were better than many others when they came out, so they were successful and became very widespread: now their "appeal" is only based on either nostalgia, or "getting used to them", or "not knowing the better alternatives", because overall they are very very poor games (especially Monopoly).
In general boardgames have a better replayability value simply because they are designed to have it, as opposite to some videogames that are (more or less) designed to be played just once. You will notice how PvP videogames usually have an higher replayability value than the PvE one: that's because playing against your friends makes the game more unpredictable than playing it against fixed opponents/levels. Since boardgames are almost exclusively PvP...
It's worth notice that both the videogame and the boardgame industry are recently experimenting different kinds of gameplay, so what I just said is only valid up to "a few years ago": now things are changing.
Something I find that helps tremendously in replayability is variable start-up. This is critical to games like Settlers of Catan and Dominion. Every time you start things are going to be very different and require a different strategy. Compare this to something like monopoly, which is going to start exactly the same and have the exact same strategies almost every time. If you've never played Dominion with a certain kingdom card, or never played Catan where you end up without any brick, its going to feel very different and re-engage you possibly as much as the first time you played it. New strategies and evaluations of new situations make up most of the difference between the game you end up playing 5-15 times and the game you end up playing 50+ times.
How this variability in strategy is achieved varies as widely as board games themselves do. Dominion has different kingdom cards, Catan has variable islands, bridge has your starting hand, Carcassonne has what tiles you draw, werewolf has different personalities you're playing with, and many cooperative games have different player roles and varying threats that need to be addressed. But the more similar a game is each time you play it, the less you'll tend to want to play it again.
From the standpoint of combinatorial games, it has to do with 3 factors"
The rules must be few so that the games are easy to learn. The gameboard and play pieces must be simple so that people can make their own sets when formal sets are not available. (A possible tic-tac-toe board, carved in stone, is at least 40,000 years old.) These games are categorized as "abstract", and and the longest-lived games in existence. [See "Elegance in Game Design" by Cameron Browne for further exploration of this topic.]
This relates to the inability of the player to solve the game. Triviality is subjective--to the average 5 year-old, tic-tac-toe is distinctly non-trivial! For the most part, games regarded as non-trivial are those they defy solution by the strongest human players, and more recently, automata. [Note that while AlphaGo can beat the strongest human players, the game itself is unlikely to be solved.] Non-triviality means the game has depth and unlimited potential replayability.
- Emergent Complexity
Emergence is a broad category, and in the context of games the term "emergent complexity" may be more apropos. (Unfortunately, the linked "emergent gameplay" wiki focuses on video games, but this quality is a factor in combinatorial games going back at least 2500 years via Go.) Emergent complexity a key reason Go and Chess have been continuously played for millennia, and relates to depth, but in the context of strategy. The basic idea is that unsolvable games are chaotic systems and new strategies will continue to emerge, which is what makes the games so compelling and long-lived.
Note: I focused on abstract, combinatorial "playgames" because they are the oldest and arguably most widely played games in existence (Chess and Tic-tac-toe in particular.) Other very old games include Backgammon, which is in a slightly different class per it utilization of randomness. Although games like Monopoly and Risk are sure to be played for many decades to come, and might last centuries, I'd be very surprised if many modern games are still widely played in 1000 years, with the exception of Tetris*, but have little doubt that the abstract, combinatorial games will still be going strong.
One of the key conditions that limits the lifespan of video games is advances in computing (processor speed, memory, graphics) which results in popular games being supplanted on an ongoing basis by newer games which utilize stronger computing capability. What makes Tetris so unique, and the reason it is potentially the most successful computer game of all time, is that it is a fundamental, abstract game, not subject to obsolescence as a result of stronger computing.
I think this simply comes down to the game mechanics.
What makes a great game great, is that depending on the strategy employed by the person you are playing against, it can significantly change the outcome of the game. Not one strategy is going to last you through winning at all times, but requires you to shift and change your strategy based on the approach your opponents are playing. A good game mechanic can ensure that the available strategy for winning is varied and dynamic.
Predictability is boring, variety is not.
That said, I am a much bigger fan of luckless games. Whilst I like to try to outsmart my opponents, I hate to lose or win by the luck of the dice or the random turn of a card.
I think the main (though not sole) factor is the competition, and depth is another major one.
I love video games, but in most of them I am playing against the computer, often with a difficulty slider, often with the ability to adjust it after the game has begun. It is fun, but not competitive. I might play it through twice (especially if a different play through style) but more than that is very rare and by then I've seen it most of everything there is to see. Enslaved: Journey to the West is one of my favorite recent vide games, but I only played it through once because there wasn't much more to see or do on a second playthrough.
With Chess or Go on the other hand, I am competing against another human (I do play against computers on both, but I see that largely as training/practice for playing against humans). They also have enormous depth, where I frequently discover things that are new to me in games or in books about the games. In contrast, while I certainly missed some things, I found most of what was available in Enslaved: Journey to the West in my playthrough. Even more, professional Go and Chess players are discovering things that are new to the entire world.
To agree with and expand on what Lohoris said, video games that incorporate depth and competition last a lot longer than others. Starcraft, which is all about competition and has enormous depth, is still played many years later and only really started dropping in popularity when Starcraft II came out with those elements and better graphics.
The social aspect is huge, and it's importance cannot be overstated. Even with all the new technology, playing video games with people in other locations doesn't provide anything close to the same camaraderie of a bunch of people in one room. And even now there are very few video games that work for more than 4 people at once, and many multiplayer video games have a steep skill curve and will be less fun if some of the participants aren't as good as others.
Another factor (inspired by the comments about Monopoly) is adaptability. One of our most popular tags here is one that's completely absent from gaming.stackexchange: [house-rules]. With board games, it's easy to experiment, try something unusual, twist and combine. Most games I like I've at least tried to play in a different way, and some of the variations have stuck. This is so easy to do with board games, and takes a lot of effort and some specialized skills to do with most computer games.
For our purposes, there are two main styles of board games: European (or German style) and American style. In recent years, the European style has become more popular because they emphasize skill rather than luck. That is, they involve primarily the earning of "income" of one sort or another, and the deployment of the resulting resources. The luck regarding "conflict" between opposing resources is a relatively minor part of the game.
The more popular American games, such as Monopoly and Risk probably have relatively greater appeal because they employ European style dynamics, specifically regarding "income" and the management of such income (long before the European style games came into vogue). In this regard, they are different from typical American style games such as those of e.g. Avalon Hill that are based on fixed resource endowments (based on historical availabilities), and direct conflict between those resources, without prior income management.