Fun is not the first thing that a product manager thinks about when designing a product. Especially those of us in enterprise software market. People think that making something fun would only apply to companies like Zynga, Electronic Arts, etc. But I think we all want a fun product as consumers, and we are all underestimating the effect of having fun when we use the product.
A classic example of a fun product is computer games. It is purposefully created to deliver 'fun' to users. So if we analyse what games deliver to their players, we must be getting closer to definition of 'fun'.
There are many wonderful talks available on YouTube about introducing game mechanics to your product, so called gamification. I recommend talks by Amy Jo Kim, Gabe Zichermann, and Jane McGonigal.
Let me put them in my own words. A fun product does following things: