Sometimes, the term is used for a particular computational method for transverse Mercator: that is, how to convert between latitude/longitude and projected coordinates. There is no simple closed formula to do so when the earth is modelled as an ellipsoid. But the Gauss–Krüger method gives the same results as other methods, at least if you are sufficiently near the central meridian: less than 100 degrees of longitude, say. Farther away, some methods become inaccurate.
The term is also used for a particular set of transverse Mercator projections used in narrow zones in Europe and South America, at least in Germany, Turkey, Austria, Slovenia, Croatia, Macedonia, Finland and Argentina. This Gauss–Krüger system is similar to the universal transverse Mercator system, but the central meridians of the Gauss–Krüger zones are only 3° apart, as opposed to 6° in UTM. As a consequence, the scale variation within a Gauss–Krüger zone is about 1/6 of what it is in a UTM zone. More information on this system can be found on the German Wikipedia.