Geshe (dge bshes). 1) A spiritual teacher according to the Mahayana teachings. 2) A learned teacher according to the Kadam and Gelug traditions.
In the Kadam tradition this is considered to be a spiritual friend that is teaching someone dharma.
In the Gelug tradition this is the title given to someone who has finished a corresponding Ph.D level spiritual education at one of the current main three monasteries, Drepung, Sera or Ganden.