A Godfather is a role that investigates as Innocent (or some other favorable result) to Cops, regardless of their actual alignment.
Godfather is a name given exclusively to Mafia members. Serial Killers with investigation immunity are simply called "investigation-immune".
Godfather was considered Normal on mafiascum.net until an update in April 2016, when it was added to the "explicitly non-Normal" list.
In games with multiple killers, instead of appearing innocent on Cop reports, the Godfather may be Bulletproof instead; they will appear guilty on Cop reports as normal, but cannot be killed at Night. (This has also been called Mafia Tough Guy.)
Some games offer Godfathers both investigation- and kill-immunity simultaneously.
In more traditional games, the Godfather may be the final arbiter in who the Mafia kills that Night, and who is performing the kill. Alternatively, the Godfather may be the only Mafioso who can kill.
An investigation-immune Godfather may lose his investigation immunity if he performs a kill himself.
Alternatively to the above, some moderators take the concept of investigation immunity to make Godfather immune to some other investigation role instead of Cop.
In the setup Vengeful Mafia, the name is used for a role that causes the Mafia to lose when lynched; this variant of the role never caught on under the name "Godfather", and has since been given the less ambiguous name Flagbearer.
Use and Balance
As with Millers, the foremost use of Godfathers is to weaken the power of a Cop. A Godfather in the setup forces the Town to lynch at least one scum without following a Cop report. In games that should otherwise be broken by process of elimination using Cop results, this is quite a boon.
Similarly, a Bulletproof Godfather ensures that a Mafia team cannot lose solely by being killed at Night; the Godfather must be lynched for them to lose. This is especially true if the Godfather is targeted to die by a Serial Killer, who cannot simply claim that their kill inexplicably failed.
The actual power of this role is not particularly high, given the odds of a particular Mafioso being investigated or killed being somewhat low in general. However, Mafias do not need much power to combat Town power roles, so a Godfather is usually sufficient in setups where Town's power is concentrated in a Cop or Vig.
An investigation immune Godfather may wish to play scummier than normal, in the hopes of getting investigated by a Cop; however, this runs the risk of being lynched for playing scummy. Alternatively, lurking early may draw a Cop investigation, because it makes you harder to read; however, this runs the risk of being lynched for lurking, especially if you are active in other games or it is contradictory to your normal playstyle.