For meat to be kosher, it must come from a kosher animal and be slaughtered in a kosher way. For meat to be glatt kosher, in addition to the two above conditions, the meat must also come from an animal with adhesion-free or smooth lungs.
The word glatt means smooth in Yiddish. In Jewish Law, the term glatt is used to refer to the lungs of animals. After the animal is slaughtered, the animal is opened and examined to determine whether the lungs are smooth. If defects on the lungs are found, the meat is considered treif (torn, mortally injured, non-kosher). If the lungs are found to be defect-free or smooth, the meat is considered to be glatt kosher.
While the term glatt technically means the lungs of the kosher and kosher-slaughtered animal were smooth, the term is often used colloquially to imply a higher standard of kashrut, similar to the term mehadrin.
Furthermore, even though only meat can be technically glatt kosher, the term is often loosely used today to refer to non-meat items. Many suppliers of glatt kosher items will refer to all their products at glatt kosher. So one may find fish with the same glatt kosher sticker as used on meat being sold one aisle over. In addition, many suppliers of glatt kosher meat will refer to their whole service as glatt kosher. So there are glatt kosher caterers, restaurants and stores.