A fan can come from many things. I will try to list as many as I can to help you understand the meaning. In three fan there are some basic rules to the game.
1. Must play all one suit if you chow. You CANNOT mix chows of different suits. This is called chicken or Pung Wu. It is a cheap way of winning a hand, is frowned upon and will get you booted from the game.
2. An all Pong hand can be of different suits. But, if you mix your suits, you lose points in your final hand.
You can add Honors to your pong hand, and they will add points, but better if they are your position wind, prevailing wind, or dragon of your suit.
By dragon of your suit, I mean that the dragon honors match the different suits that you may choose to play. And you must play one suit only.
Green Dragon
+ Bamboo
White Dragon
+ Dots
3. Make sure you have three of the following before you Mahjong:
Position
Wind - Pong or Kong (Honors)
Dragon
- Pong or Kong (Honors)
Terminal
Pong or Kong <----(This is under debate)
>^..^<
Season
of correct number (seat position)
Flower
of correct number (seat position)