There are countless posts about fun by now. And as long as I think that I make progress at pinning it down, or at least revealing interesting characteristics, I'll probably continue writing about it. That having said, let's jump in.
Humans could do all kinds of stuff with their mental and physical capabilities. We could solve differential equations all day or repeat singing the same song for hours. What we do or don't do is decided in our brains and mostly happens on a subconscious level.
Since our brains evolved to maximize the probability of reproduction of our immediate social group (natural selection), they usually make us do things that help with that 'goal'. Fun is a tool the brains use to this end.
The question, “What is fun?”, can now be answered: "Fun is what our brain makes us feel, if it 'wants' us to do something." Humans are sophisticated robots in this model.
While we engage in an activity our brains constantly check whether the activity is still 'worthwhile' to the end that it maximizes the chance of reproduction of our immediate social group. But this is a pretty damn hard choice to make because our brains miss lots of information. And thus they can quite easily be fooled.
The job of a game designer is to fool the brains of other humans into thinking that an activity, which may be pointless, is actually worth doing because it is fun.
If you like to play games you want to be fooled. There's nothing wrong with this. I'm sure you already realized that playing video games doesn't maximize the chance of reproduction of your immediate social group.