Flavor is fluff material included in many Mafia games to add a story or plot, or to connect themed games to their source material.
Moderators include flavor to enhance role PMs and make them more interesting, to set up execution and nightkill scenes, and to explain how night actions work. Detailed flavor can often make the game far more interesting for players and moderator alike, and can even provide a roleplaying element to a game if a player so chooses. Likewise flavor can also be used to turn mafia from simply a game into a collaborative storytelling experience, which is pretty cool. However it is important to be careful with flavor, because misuse can cause players to make flavor arguments.
Some uses of flavor can tip the balance of a game towards the town or scum beyond simply the roles, skill, or luck of the players. For example, in themed games, it can be very easy for players to get confirmed by nature of the character they are playing, since many moderators keep central characters for the town to use. For example, in a Star Wars game, a "Luke Skywalker" claim could be enough to clear someone as town. This can be countered by giving scum fakeclaims, like in an X-Men game making Professor Xavier scum.
Some flavor can also confirm players through night actions by revealing more information than the role "should" know. For example, if a Cop receives flavor in a flavor-heavy game that he visits his target's house but finds the door already open, implying the target has a night action, then the Cop has discovered more information than his role would typically allow him to get. In this way, flavor can be used to confirm the night actions of players that would be impossible to confirm in other games.
Finally, flavor (especially PM flavor) can be used to confirm or condemn players, especially in games with multiple roles. One of the standard mafia rules is that players may not quote their exact PMs, but even through paraphrasing, players can manage to confirm each other based on the similarity of their flavor. This is especially a problem for vanilla townies, as moderators generally use the same Role PMs for all vanilla players.
How To Use Flavor Well
There are a few ways to prevent flavor from affecting balance. The easiest (but least interesting) is to simply have no flavor at all, or keeping it minimal enough that players cannot use it for good or for ill. Another popular way to balance flavor is to allow the mafia access to a "flavor-proof" fakeclaim, either from the moderator writing them fake, town role PMs, or giving them (or even publishing in the OP) the vanilla role PM if applicable. More experienced moderators can even play with flavor, perhaps by making flavor unreliable, or writing flavor anticipating players' reactions and responses, or even making different players' flavor deliberately contradict each other.
However most players in the SA Mafia community know not to take flavor too seriously. So long as you take measures to make it not unbalance a game, flavor is fine and fun.