To put it briefly, it depends on the context of the game and the reasons people are playing it. While on the surface, the original Doom appears similar to Quake 3, they are polar opposite games from a game design perspective. Doom is all about mowing downs armies of generally stupid demons with your increasingly diverse stockpile of weapons. Quake 3 is about fighting a skilled opponent with access to the same weapons and powerups you can use. Doom is a visceral game while Quake 3 is tactical. Rather than saying Doom is fun, it's more accurate to say that visceral experiences are fun, and Doom happens to fill that role. But a game like Serious Sam is just as fun, and in the same way.
If games are candy for the mind, then whether or not you like a particular flavor of game is a question of personal preference. What does your mind enjoy? Each kind of thinking corresponds to a different style of game.
Examples: Chess, Go
Requires: Discrete set of Options
People who like analytical games approach them as a puzzle to be solved. The game needs to provide decisions for the player to make, any of which could be helpful in the right situation. The fun is in selecting your decision and finding out if it was a good choice. Winning the game usually means you were right, or at least more right than your opponents. Analytical games can also be single player games, such as a Rubix Cube. They need not include perfect information to all players like Chess and Go do, of course. A game like Poker contains a lot of analysis. But the analysis is in large part about probabilities, risks, and rewards. You can win a game of poker by making the wrong play, or lose making the right play, but people who deduce the right plays more often will win more often, and that's what makes the game analytical.
Examples: Golf, Football
Requires: Physical Interaction
An action game is anything that involves a physical component. If there is ever a game situation where you know what you need to do but your muscles might not be able to do it, it's an action game. The joy in an action game is seeing how well you can do what you tried to do. It's really satisfying to shoot a three point shot in basketball, score a goal in football, or hit a home run in baseball. Every physical sport is an action game by this definition. But other games have action components as well. All first person shooter games are "action games" because they require you to aim at your target and shoot them. You know what you want to do, but your muscles might fail at moving the mouse to the right place. Even a game like Jenga is an action game, because it depends on precise muscle control.
Examples: Final Fantasy VII, World of Warcraft
Discovery games are fun because the player experience new things. Their joy is the joy of appreciation. A common theme in these games is a story line. As a result, discovery games generally have lower replayability. They are highly engaging the first time through, however. What the player experiences doesn't have to be the story, of course. It could be enjoying the look and feel of a new World of Warcraft zone. What all discovery games have in common is artistically designed content that the player enjoys experiencing.
Examples: Charades, Magic: The Gathering
Requires: Very large set of Options
Some people really enjoy thinking outside the box. To become a medium for expressing creativity, the rules of the game must be extremely flexible, allowing for a wide variety of possible options. The number of options is far larger than that of an analytical game: it needs to be so large that players cannot exhaustively try all of the alternatives. This gives creative people the option to express themselves by making choices other people might not even have considered. Creative games often have a verbal component to them, but this is not necessary. For example, deck design in Magic: The Gathering is a very creative endeavor. There are more legal Magic decks than atoms in the Universe! And a number of Magic players enjoy making decks more than playing with them.
Examples: Mafia, Shadows over Camelot
While it's true that games are only as fun as the people you play them with, some people explicitly enjoy games because of the social interactions. In their opinion, a game's purpose is to generate social interactions, and whether you win or lose isn't as important as how you got there. These games categorically encourage players to work as a team towards a common goal. Victories and losses are shared. The fun is about being part of a group and, for some games, deducing who isn't really a member of the team from their actions.
Examples: Counterstrike, Team Fortress 2
Requires: Winners and Losers
There are some gamers who play games to win. Obviously the typical sore loser falls into this category, but that's not the only type of person who fits this description. This type of gamer considers winning an expression of who they are. The sore loser is someone who considers that expression to be a validation of identity-- they are distressed when they lose because the loss injures their ego. By definition, every game has winners and losers. But in almost all cases, gamers who want to win prefer games against human opponents, as it makes winning more meaningful. Note that these gamers need not be good players. Often bad players who want to win will gravitate towards teamplay games like Counterstrike. No matter how bad you are, you will win roughly half of your games if the teams are large enough. When a gamer wants to express themselves through winning, all they really need is a human opponent and the plausibility to claim they won through skill. Actual skill could have been involved, but it's not required.