The fundamental fairness doctrine is an alternative to the doctrine of incorporation. This doctrine holds that the 14th Amendment does not hold the states to the provisions of the Bill of Rights. Instead, it holds that the states must treat citizens in a way that is fair.
A major argument for this doctrine is that it can be more comprehensive and fairer than the incorporation doctrine. It can be more comprehensive in that it can hold the states to a higher standard. If something is not in the Bill of Rights, it may still be required of the states because it is necessary for "fundamental fairness." Conversely, if there is something in the Bill of Rights that is not necessary for fundamental fairness (like the requirement for a grand jury, which the Court held was not necessary in Hurtado v. California) then states will not be held to that.
A major argument against the doctrine is that it does not adhere as closely to the actual words of the Constitution. It makes sense to define fairness with reference to the Constitution itself. If we adhere to the more nebulous idea that the states must do whatever is "fundamentally fair" we open ourselves up to judicial activism. We give judges too much discretion to decide what is and is not fair.