“Glatt Kosher” means something like “extra kosher” and applies to chicken and fish as well as meat.
Glatt is Yiddish for smooth, and in the context of kashrut it means that the lungs of the animal were smooth, without any adhesions that could potentially prohibit the animal as a trefa, an issue only applicable to animals, not fowl or nonmeat products.
In colloquial discourse, tref refers to anything that is not kosher. The technical definition of trefa is based on Exodus 22:30 (“Do not eat meat from an animal torn [trefa] in the field”) and refers to an animal with any of a specific group of physical defects that are detailed in the [Babylonian] Talmud (most of the third chapter of Hullin; 42a-59a) and [traditional Jewish law] codes (Maimonides, Mishneh Torah, Laws of Forbidden Foods 4:6-9 and Laws of Slaughter ch. 5 -11; Shulchan Arukh, Yoreh De’ah [“YD”] 29-60). Examples of these “defects,” which often go far beyond the health inspection of the USDA, include certain lesions, lacerations, broken limbs, missing or punctured organs, or the result of an attack by a larger animal. Such defects can occur in, and thereby render, both animals and fowl tref. Because most of these defects are uncommon, it may be assumed that most animals are healthy (Shach [commentary to Shulchan Arukh], YD 39:1) and hence there is no requirement to inspect every animal for them. (This does not imply that a blind eye may be turned to their presence.)
Lungs are a special concern
An exception is the lung of an animal, on which adhesions [sirhot] and other problems may develop. While these problems are not common, they do occur more frequently than other trefot [i.e., defects]. Their relative prevalence led the rabbis to mandate that the lungs of every animal be examined, both manually while still in its natural position in the animal, and visually following its removal from the thoracic cavity (Shulchan Arukh, YD 39:1). (Nowadays, another problem that occurs with relative frequency and is therefore also inspected for is holes of the second stomach, the bet ha-kosot [reticulum], caused by animals eating nails and other sharp metal objects.) Because a hole in the lung renders the animal a trefa, adhesions, i.e., pathologically arising bands of collagen fibers, are problematic either because they indicate the presence of a perforation that has been insufficiently sealed ([so says Talmud commentator] Rashi) or because they can become loosened, thereby causing a hole to develop ([so say the Talmud commentators known collectively as] Tosafot).