A stinking gizzard is a drinking establishment (that may or may not serve food items) that notoriously elicits derogatory remarks from customers pertaining to a characteristic stench that most often emanates from the male and female bathrooms (if there is a distinction) with the stench often consisting of aromas most often associated with urine, feces, vomit, sex, and regurgitated food.
Note that the noun phrase "stinking gizzard" may be preceded by the definite and indefinite determiners ('the' and 'a' respectively). "The stinking gizzard" is often used when there is a particularly salient drinking establishment that is notoriously 'a stinking gizzard'.
Hey man, we're going out to the stinking gizzard tonight. The place is rancid but at least the beer is cheap.