Every game designer, analyst, and reviewer has to mention this somewhere. I'm sure that almost everything that needs said about this has been already, but everyone still expects me to talk about it anyway. This also has the plus side of offering those of you interested in game design but unfamiliar with this particular idea a kind of crash course to get you stared.
The process that led up to the decision to write about this now deserves some mention. A few weeks ago I wrote an article about trying to make gameplay feel important(or at least progressive toward a certain goal) to the player. I misrepresented that concept as being an essential design part of every game. Someone mentioned in the comments that not all games need this, and gave old arcade games like PacMan as an example. While I ultimately find PacMan(and a lot of those old games) get boring to me very fast, it still got in my head to talk about why people wouldn't get bored. After all, many people have spent hours and hours on PacMan, and there is really no solid story to the game in the first place.
This leads up to a noted phenomenon(which might even be named) in game design. It's basically the observation that different people in different moods enjoy different things about games at different times. For example, I myself don't always give a crack about whether or not what I'm doing makes me feel like I'm making progress, or even making any difference at all. Sometimes I just want to kick back, relax, and make things explode. And that's just me, tastes vary between people as well. Let's look at a few of these things people enjoy about games.
I myself tend to lean more toward games that have an emphasis on story, like Deus Ex: Human Revolution. As I've mentioned in the past, I usually get bored of interesting gameplay mechanics a little faster than most. But if these interesting mechanics are coupled with an interesting story full of characters I enjoy, one that keeps me pushing ahead to see what happens next, I can easily play for hours on end. Observation leads me to believe I'm not the only one like this either, so story is definitely on the list.
Now while I love a good story, sometimes I don't even posses the mental stamina to pay attention to one even if I wanted to. I might be under stress, sick of dealing with inter-personal relationships, tired, and just wanting to see things burn. Then I would probably want to play a fast paced, frantic shooting game like GridWars. A game where I could just forget about the fact that there is no story, no reason to be here shooting up glowing vector enemies, and no way to win. A game where I can just have some fun, smooth playing, shooting. This speaks more to the primal instincts of human nature. It can be a chance for catharsis, or it can just be naturally enjoyable. Everyone has human nature, so cathartic and naturally "fun" mechanics are also on the list.
Yet another common reason one might enjoy a game is for the escape. Sometimes a game is fun just because it allows you to enter a universe other than the one you're stuck living in. I'm a very cynical and pessimistic person, so I myself occasionally enjoy games for this reason. It gives me a chance to unrealistically believe that there's a shred of hope for human nature or even this planet in general. Most people really aren't satisfied with reality anyway, making this one of the top contributors.
Still, almost everyone enjoys puzzle games from time to time. I myself like to indulge in a good game of chess with my friends every once in awhile, or maybe I play some Portal when I'm in the mood. This speaks to a few different core concepts people enjoy. One of them is the mere intellectual stimulation. I personally love logical reasoning and critical thinking. A game which requires these tickles my brain. But puzzle games often also speak to your feeling of accomplishment. When you spend a whole hour figuring out this one puzzle, cleverly coming up with out-of-the-box ways around problems, it makes you feel awesome to see it finished. You look back at what you've accomplished and revel in your adeptness, even when it doesn't exist.
Puzzle games and difficult arcade games alike also challenge the player. Some people(myself included) just inherently enjoy being presented with something challenging yet theoretically possible. If you succeed it can speak to your feeling of accomplishment, but even if you don't you may still enjoy having tried. A lot of hackers are this way: trying to break into a system just because it looks like it would be fun.
Essentially, there are a ton of different aspects of games people can enjoy. We have story, characters, cathartic and "fun" mechanics, escape, feeling of accomplishment, intellectual stimulation, challenge and more. I haven't even mentioned aesthetics, socialisation, art, etc. This is a very important concept to keep in mind as a designer. You need to know what about your game is going to appeal to humans at the base level. This will also help you predict and choose your audience.
To wrap up and go full circle, lets go back to PacMan. A game doesn't need to have everything people could enjoy in it. Some games are all action and gameplay with no story. Some games are all story with pretty terrible gameplay. PacMan is an example of a game that appeals mainly through challenge and score eliteness. Thus, it doesn't need to have a clever story which the player interactively experiences.