A geshe is a scholar who has completed the long monastic curriculum of the Gelug school of Tibetan Buddhism, a course of study that classically runs for many years.
The degree of learning
The title marks the completion of the extended monastic course of study of the Gelug school, classically some fifteen to twenty years or more of training in logic, epistemology, the perfection of wisdom, the middle way philosophy, monastic discipline and metaphysics, pursued through the rigorous method of formalised debate, so the geshe is above all a master of the scholastic and philosophical tradition. The highest ranks of the degree are awarded to those who excel in this long course and its examinations, and the geshe is qualified to teach the philosophical curriculum.
Learning, not tantric office
The geshe standing is scholarly, and it is distinct from the office of the lama who confers tantric empowerment and from recognition as a reincarnate teacher, so a geshe is a learned scholar-monk who may or may not also hold those other roles, and the conflation of scholarly and tantric authority is an error the distinction guards against. The Sakya school has a comparable scholarly qualification. Entry is by the completion of the curriculum within the great monastic colleges, so the office rests on demonstrated mastery of the philosophical tradition. The geshe thus represents the scholarly summit of the Gelug system, the master of its long course of study and debate, the counterpart in learning to the offices of empowerment and recognition that carry other kinds of authority in the tradition.