Gen (short for "general") usually means a Fan Fic that includes no pairing, be they homo-erotic or hetero-erotic.
However, there are some disagreements as to what exactly that means.
To most everyone it includes fics without any smut, but to some it goes as far as to mean a fic where not the slightest hint of a romantic relationship shows up.
For some people it includes a fic that isn't focused on the romance, but where pairings can feature in the background. Some people make a difference between showing canonical pairings and non canonical pairings (with all the debates that can lead to about what's a canonical pairing), which many slashers criticize as being hetero-sexist, because the majority of what's acknowledged as canonical pairings do not include their ships.
It's sometimes used to mean an Original Flavour fic.
Beyond what it doesn't include, Gen's a category for plot-focused stories or light day-in-the-life stories. It's often popular among fans who like to explore all the dynamics of the ensemble cast, to put the focus more on the family and the friendly relationships.
Gen fans often complain of being the red-headed stepchild of ficdom, and indeed, it's much less popular and more difficult to find than any fic focused on Shipping.
Usage of this term tends to be limited to ficdom where Shipping Fic makes up the bulk of the material produced (which are the most common).