There are only two types of games: puzzle and skill games. Though most games are actually hybrids, usually one of the types is dominant.
For example, single-player action games, strategy games, and adventure games are essentially just puzzles. Once you figure out, what you have to do, they are easy. In contrast, competitive multiplayer games are usually about skill. Everybody knows what to do in Quake, but the hard part is to perform under stress. Jump-n-run games are often hybrids. At some parts you just need to find the button (puzzle) and next you have to get the platform jumping right (skill).
This classification easily extends to non-computer games. For example, sports are all about skill. Most board games are about skill, as well. Pen-and-paper roleplay is about solving the puzzles the game master comes up with. Poker is also about skill (though less about bluffing and more about probabilities than most people think).
In addition there is the story telling aspect, which is not game-specific, though. Stories in itself are thrilling. Just think of books and movies. Games are about interaction, though. Nonetheless, games are a great medium to tell stories (and not just with cut scenes).
Now that we have this classification (puzzle, skill, hybrid), let us look at some tricky cases.
What are those social games (World of Warcraft, Farmville, etc) about? The initial puzzling (how to play?) wears off fast. The skill does not make much of a difference. What distinguishes between good and bad players is mostly the effort/time they put in. The hardcore players usually talk about the community spirit. Effectively, I would classify those "social games" as socializing activities, but not as games. Likewise, all games based on luck (Roulette, Ludo, etc) are more about the socializing than the gaming.
Now what about those "sandbox games" like Minecraft? Like social games, the skill and puzzle parts are relatively small. It is about building cool things, which I would label as telling stories. So, Minecraft is actually more about storytelling than gaming.
Finally, let me clarify that I do not want to value games with this classification. Whether something is fun, is a completely different question. This article just carves out, what a game really is.